A334254_and_A334255.ipynb

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   "source": [
    "# Computation of OEIS sequences A334254 and A334255 for n=6 and checking their values for n=3,4,5. "
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    "# 1. Main code."
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       "static PyObject *__pyx_f_46_cython_magic_a771caaea75b905f51e41df3c6438720_ProcessRT(PyObject *__pyx_v_t, int __pyx_v_i, int __pyx_v_l, int __pyx_v_nn1, CYTHON_UNUSED int __pyx_skip_dispatch) {\n",
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       "static char __pyx_doc_46_cython_magic_a771caaea75b905f51e41df3c6438720_ProcessRT[] = \" This function recursively extends an input tuple of integers, which represents \\n    a set of sets (more precisily, a formal context), by adding to it an integer i (a new set).\\n    The length of a terminal tuple is constrained by l, while nn1 should be passed as 2**n-1. \\n    The function checks whether the new tuple forms a unique closure system and returns the dictionary of the first elements of\\n    processed tuples as keys and values 1 for clousre system with T1 propery and 0 otherwise.\";\n",
       "static PyObject *__pyx_pw_46_cython_magic_a771caaea75b905f51e41df3c6438720_1ProcessRT(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
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       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">43</span>:         <span class=\"n\">r</span><span class=\"o\">=</span><span class=\"mf\">0</span></pre>\n",
       "<pre class='cython code score-0 '>    __pyx_v_r = 0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">44</span>:         <span class=\"k\">for</span> <span class=\"n\">k</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"o\">+</span><span class=\"mf\">1</span><span class=\"p\">,</span><span class=\"n\">nn1</span><span class=\"p\">):</span></pre>\n",
       "<pre class='cython code score-0 '>    __pyx_t_12 = __pyx_v_nn1;\n",
       "    __pyx_t_14 = __pyx_t_12;\n",
       "    for (__pyx_t_15 = (__pyx_v_i + 1); __pyx_t_15 &lt; __pyx_t_14; __pyx_t_15+=1) {\n",
       "      __pyx_v_k = __pyx_t_15;\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">45</span>:             <span class=\"c\">#print(&quot;Process(t,k,l-1,tuples,nn1)&quot;)</span></pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">46</span>:             <span class=\"c\">#print(&quot;t&quot;,t,&quot;k&quot;,k)</span></pre>\n",
       "<pre class=\"cython line score-1\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">47</span>:             <span class=\"n\">ucount</span><span class=\"o\">=</span><span class=\"n\">ProcessRT</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">,</span><span class=\"n\">k</span><span class=\"p\">,</span><span class=\"n\">l</span><span class=\"o\">-</span><span class=\"mf\">1</span><span class=\"p\">,</span><span class=\"n\">nn1</span><span class=\"p\">)</span></pre>\n",
       "<pre class='cython code score-1 '>      __pyx_t_3 = __pyx_f_46_cython_magic_a771caaea75b905f51e41df3c6438720_ProcessRT(__pyx_v_t, __pyx_v_k, (__pyx_v_l - 1), __pyx_v_nn1, 0);<span class='error_goto'> if (unlikely(!__pyx_t_3)) __PYX_ERR(0, 47, __pyx_L1_error)</span>\n",
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       "      __pyx_t_3 = 0;\n",
       "</pre><pre class=\"cython line score-43\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">48</span>:             <span class=\"k\">for</span> <span class=\"n\">key</span> <span class=\"ow\">in</span> <span class=\"n\">ucount</span><span class=\"p\">:</span></pre>\n",
       "<pre class='cython code score-43 '>      if (likely(<span class='py_c_api'>PyList_CheckExact</span>(__pyx_v_ucount)) || <span class='py_c_api'>PyTuple_CheckExact</span>(__pyx_v_ucount)) {\n",
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       "</pre><pre class=\"cython line score-2\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">49</span>:                 <span class=\"k\">if</span> <span class=\"n\">key</span> <span class=\"ow\">in</span> <span class=\"n\">count</span><span class=\"p\">:</span></pre>\n",
       "<pre class='cython code score-2 '>        __pyx_t_8 = (<span class='pyx_c_api'>__Pyx_PyDict_ContainsTF</span>(__pyx_v_key, __pyx_v_count, Py_EQ)); if (unlikely(__pyx_t_8 &lt; 0)) <span class='error_goto'>__PYX_ERR(0, 49, __pyx_L1_error)</span>\n",
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       "/* … */\n",
       "          goto __pyx_L17;\n",
       "        }\n",
       "</pre><pre class=\"cython line score-19\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">50</span>:                     <span class=\"n\">count</span><span class=\"p\">[</span><span class=\"n\">key</span><span class=\"p\">]</span><span class=\"o\">+=</span><span class=\"n\">ucount</span><span class=\"p\">[</span><span class=\"n\">key</span><span class=\"p\">]</span></pre>\n",
       "<pre class='cython code score-19 '>          <span class='pyx_macro_api'>__Pyx_INCREF</span>(__pyx_v_key);\n",
       "          __pyx_t_1 = __pyx_v_key;\n",
       "          __pyx_t_11 = <span class='pyx_c_api'>__Pyx_PyDict_GetItem</span>(__pyx_v_count, __pyx_t_1);<span class='error_goto'> if (unlikely(!__pyx_t_11)) __PYX_ERR(0, 50, __pyx_L1_error)</span>\n",
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       "          if (unlikely(<span class='py_c_api'>PyDict_SetItem</span>(__pyx_v_count, __pyx_t_1, __pyx_t_5) &lt; 0)) <span class='error_goto'>__PYX_ERR(0, 50, __pyx_L1_error)</span>\n",
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       "          <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_1); __pyx_t_1 = 0;\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">51</span>:                 <span class=\"k\">else</span><span class=\"p\">:</span></pre>\n",
       "<pre class=\"cython line score-8\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">52</span>:                     <span class=\"n\">count</span><span class=\"p\">[</span><span class=\"n\">key</span><span class=\"p\">]</span><span class=\"o\">=</span><span class=\"n\">ucount</span><span class=\"p\">[</span><span class=\"n\">key</span><span class=\"p\">]</span></pre>\n",
       "<pre class='cython code score-8 '>        /*else*/ {\n",
       "          __pyx_t_1 = <span class='pyx_c_api'>__Pyx_PyObject_GetItem</span>(__pyx_v_ucount, __pyx_v_key);<span class='error_goto'> if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 52, __pyx_L1_error)</span>\n",
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       "          <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_1); __pyx_t_1 = 0;\n",
       "        }\n",
       "        __pyx_L17:;\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">53</span>:             <span class=\"c\">#if r&lt;0: print(&quot;r&quot;,r)</span></pre>\n",
       "<pre class=\"cython line score-2\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">54</span>:         <span class=\"k\">return</span> <span class=\"n\">count</span></pre>\n",
       "<pre class='cython code score-2 '>    <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_r);\n",
       "    <span class='pyx_macro_api'>__Pyx_INCREF</span>(__pyx_v_count);\n",
       "    __pyx_r = __pyx_v_count;\n",
       "    goto __pyx_L0;\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">55</span>:     <span class=\"k\">else</span><span class=\"p\">:</span></pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">56</span>:         <span class=\"c\">#print(tuples)</span></pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">57</span>:         <span class=\"c\">#tuples.append(tuple(t))</span></pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">58</span>: </pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">59</span>: </pre>\n",
       "<pre class=\"cython line score-14\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">60</span>:         <span class=\"k\">print</span><span class=\"p\">(</span><span class=\"s\">&quot;Branch process&quot;</span><span class=\"p\">,</span><span class=\"n\">t</span><span class=\"p\">[</span><span class=\"mf\">0</span><span class=\"p\">])</span></pre>\n",
       "<pre class='cython code score-14 '>  /*else*/ {\n",
       "    __pyx_t_3 = <span class='pyx_c_api'>__Pyx_GetItemInt</span>(__pyx_v_t, 0, long, 1, __Pyx_PyInt_From_long, 0, 0, 1);<span class='error_goto'> if (unlikely(!__pyx_t_3)) __PYX_ERR(0, 60, __pyx_L1_error)</span>\n",
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       "    <span class='pyx_macro_api'>__Pyx_INCREF</span>(__pyx_kp_u_Branch_process);\n",
       "    <span class='refnanny'>__Pyx_GIVEREF</span>(__pyx_kp_u_Branch_process);\n",
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       "    <span class='py_macro_api'>PyTuple_SET_ITEM</span>(__pyx_t_1, 1, __pyx_t_3);\n",
       "    __pyx_t_3 = 0;\n",
       "    __pyx_t_3 = <span class='pyx_c_api'>__Pyx_PyObject_Call</span>(__pyx_builtin_print, __pyx_t_1, NULL);<span class='error_goto'> if (unlikely(!__pyx_t_3)) __PYX_ERR(0, 60, __pyx_L1_error)</span>\n",
       "    <span class='refnanny'>__Pyx_GOTREF</span>(__pyx_t_3);\n",
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       "    <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_3); __pyx_t_3 = 0;\n",
       "</pre><pre class=\"cython line score-2\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">61</span>:         <span class=\"k\">return</span> <span class=\"n\">count</span></pre>\n",
       "<pre class='cython code score-2 '>    <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_r);\n",
       "    <span class='pyx_macro_api'>__Pyx_INCREF</span>(__pyx_v_count);\n",
       "    __pyx_r = __pyx_v_count;\n",
       "    goto __pyx_L0;\n",
       "  }\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">62</span>: </pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">63</span>: </pre>\n",
       "<pre class=\"cython line score-45\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">64</span>: <span class=\"k\">def</span> <span class=\"nf\">isClosed</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">,</span><span class=\"n\">g</span><span class=\"p\">):</span></pre>\n",
       "<pre class='cython code score-45 '>/* Python wrapper */\n",
       "static PyObject *__pyx_pw_46_cython_magic_a771caaea75b905f51e41df3c6438720_3isClosed(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds); /*proto*/\n",
       "static char __pyx_doc_46_cython_magic_a771caaea75b905f51e41df3c6438720_2isClosed[] = \"Checks whether g is closed in t.\";\n",
       "static PyMethodDef __pyx_mdef_46_cython_magic_a771caaea75b905f51e41df3c6438720_3isClosed = {\"isClosed\", (PyCFunction)(void*)(PyCFunctionWithKeywords)__pyx_pw_46_cython_magic_a771caaea75b905f51e41df3c6438720_3isClosed, METH_VARARGS|METH_KEYWORDS, __pyx_doc_46_cython_magic_a771caaea75b905f51e41df3c6438720_2isClosed};\n",
       "static PyObject *__pyx_pw_46_cython_magic_a771caaea75b905f51e41df3c6438720_3isClosed(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
       "  PyObject *__pyx_v_t = 0;\n",
       "  PyObject *__pyx_v_g = 0;\n",
       "  PyObject *__pyx_r = 0;\n",
       "  <span class='refnanny'>__Pyx_RefNannyDeclarations</span>\n",
       "  <span class='refnanny'>__Pyx_RefNannySetupContext</span>(\"isClosed (wrapper)\", 0);\n",
       "  {\n",
       "    static PyObject **__pyx_pyargnames[] = {&amp;__pyx_n_s_t,&amp;__pyx_n_s_g,0};\n",
       "    PyObject* values[2] = {0,0};\n",
       "    if (unlikely(__pyx_kwds)) {\n",
       "      Py_ssize_t kw_args;\n",
       "      const Py_ssize_t pos_args = <span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args);\n",
       "      switch (pos_args) {\n",
       "        case  2: values[1] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 1);\n",
       "        CYTHON_FALLTHROUGH;\n",
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       "        CYTHON_FALLTHROUGH;\n",
       "        case  0: break;\n",
       "        default: goto __pyx_L5_argtuple_error;\n",
       "      }\n",
       "      kw_args = <span class='py_c_api'>PyDict_Size</span>(__pyx_kwds);\n",
       "      switch (pos_args) {\n",
       "        case  0:\n",
       "        if (likely((values[0] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_t)) != 0)) kw_args--;\n",
       "        else goto __pyx_L5_argtuple_error;\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  1:\n",
       "        if (likely((values[1] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_g)) != 0)) kw_args--;\n",
       "        else {\n",
       "          <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"isClosed\", 1, 2, 2, 1); <span class='error_goto'>__PYX_ERR(0, 64, __pyx_L3_error)</span>\n",
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       "        if (unlikely(<span class='pyx_c_api'>__Pyx_ParseOptionalKeywords</span>(__pyx_kwds, __pyx_pyargnames, 0, values, pos_args, \"isClosed\") &lt; 0)) <span class='error_goto'>__PYX_ERR(0, 64, __pyx_L3_error)</span>\n",
       "      }\n",
       "    } else if (<span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args) != 2) {\n",
       "      goto __pyx_L5_argtuple_error;\n",
       "    } else {\n",
       "      values[0] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 0);\n",
       "      values[1] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 1);\n",
       "    }\n",
       "    __pyx_v_t = values[0];\n",
       "    __pyx_v_g = values[1];\n",
       "  }\n",
       "  goto __pyx_L4_argument_unpacking_done;\n",
       "  __pyx_L5_argtuple_error:;\n",
       "  <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"isClosed\", 1, 2, 2, <span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args)); <span class='error_goto'>__PYX_ERR(0, 64, __pyx_L3_error)</span>\n",
       "  __pyx_L3_error:;\n",
       "  <span class='pyx_c_api'>__Pyx_AddTraceback</span>(\"_cython_magic_a771caaea75b905f51e41df3c6438720.isClosed\", __pyx_clineno, __pyx_lineno, __pyx_filename);\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return NULL;\n",
       "  __pyx_L4_argument_unpacking_done:;\n",
       "  __pyx_r = __pyx_pf_46_cython_magic_a771caaea75b905f51e41df3c6438720_2isClosed(__pyx_self, __pyx_v_t, __pyx_v_g);\n",
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       "<pre class='cython code score-6 '>      __pyx_t_5 = <span class='py_c_api'>PyNumber_And</span>(__pyx_v_h, __pyx_v_res);<span class='error_goto'> if (unlikely(!__pyx_t_5)) __PYX_ERR(0, 73, __pyx_L1_error)</span>\n",
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       "  <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_2); __pyx_t_2 = 0;\n",
       "</pre><pre class=\"cython line score-42\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">85</span>:         <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">isClosed</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">,</span><span class=\"n\">m</span><span class=\"p\">):</span></pre>\n",
       "<pre class='cython code score-42 '>    <span class='pyx_c_api'>__Pyx_GetModuleGlobalName</span>(__pyx_t_1, __pyx_n_s_isClosed);<span class='error_goto'> if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 85, __pyx_L1_error)</span>\n",
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       "    if (CYTHON_UNPACK_METHODS &amp;&amp; unlikely(<span class='py_c_api'>PyMethod_Check</span>(__pyx_t_1))) {\n",
       "      __pyx_t_3 = <span class='py_macro_api'>PyMethod_GET_SELF</span>(__pyx_t_1);\n",
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       "        PyObject* function = <span class='py_macro_api'>PyMethod_GET_FUNCTION</span>(__pyx_t_1);\n",
       "        <span class='pyx_macro_api'>__Pyx_INCREF</span>(__pyx_t_3);\n",
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       "        <span class='pyx_macro_api'>__Pyx_DECREF_SET</span>(__pyx_t_1, function);\n",
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       "    if (<span class='py_c_api'>PyFunction_Check</span>(__pyx_t_1)) {\n",
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       "</pre><pre class=\"cython line score-3\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">86</span>:             <span class=\"k\">return</span> <span class=\"bp\">False</span></pre>\n",
       "<pre class='cython code score-3 '>      <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_r);\n",
       "      <span class='pyx_macro_api'>__Pyx_INCREF</span>(Py_False);\n",
       "      __pyx_r = Py_False;\n",
       "      <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_2); __pyx_t_2 = 0;\n",
       "      goto __pyx_L0;\n",
       "</pre><pre class=\"cython line score-2\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">87</span>:     <span class=\"k\">return</span> <span class=\"bp\">True</span></pre>\n",
       "<pre class='cython code score-2 '>  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_r);\n",
       "  <span class='pyx_macro_api'>__Pyx_INCREF</span>(Py_True);\n",
       "  __pyx_r = Py_True;\n",
       "  goto __pyx_L0;\n",
       "</pre></div></body></html>"
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      "text/plain": [
       "<IPython.core.display.HTML object>"
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     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%cython --annotate\n",
    "from math import log \n",
    "\n",
    "cpdef ProcessRT(t,int i,int l,int nn1):\n",
    "    \"\"\"This function recursively extends an input tuple of integers, which represents \n",
    "    a set of sets (more precisily, a formal context), by adding to it an integer i (a new set).\n",
    "    The length of a terminal tuple is constrained by l, while nn1 should be passed as 2**n-1. \n",
    "    The function checks whether the new tuple forms a unique closure system and returns the dictionary of the first elements of\n",
    "    processed tuples as keys and values 1 for clousre system with T1 propery and 0 otherwise.\"\"\"\n",
    "    \n",
    "    count={}\n",
    "    t=t[:]\n",
    "    cdef int P\n",
    "    cdef int lent=len(t)\n",
    "\n",
    "    \n",
    "    #checking whether the context is reducible\n",
    "    for j in range(lent-1,-1,-1):\n",
    "        P=i\n",
    "        #print(\"t[j],i\",t[j],i,t[j]&i,)\n",
    "        tji=t[j]&i\n",
    "        if tji in t[0:j]:\n",
    "            return count\n",
    "            \n",
    "        if tji==t[j]:\n",
    "            #print(\"t[j],i\",t[j],i,t[j]&i,)\n",
    "            for h in range(j+1,lent):\n",
    "                if t[j]&t[h]==t[j]:\n",
    "                    P=P&t[h]\n",
    "                    if P==t[j]:                        \n",
    "                        return count\n",
    "    t.append(i)\n",
    "\n",
    "    #checking T1 property\n",
    "    if Check(t,nn1):\n",
    "        count[(lent+1,t[0])]=1\n",
    "    else:\n",
    "        count[(lent+1,t[0])]=0\n",
    "\n",
    "    #recursive call\n",
    "    cdef unsigned long long r\n",
    "    #print(l)\n",
    "    if l!=0:\n",
    "        r=0\n",
    "        for k in range(i+1,nn1):\n",
    "            #print(\"Process(t,k,l-1,tuples,nn1)\")\n",
    "            #print(\"t\",t,\"k\",k)\n",
    "            ucount=ProcessRT(t,k,l-1,nn1)\n",
    "            for key in ucount:\n",
    "                if key in count:\n",
    "                    count[key]+=ucount[key]\n",
    "                else:\n",
    "                    count[key]=ucount[key]\n",
    "            #if r<0: print(\"r\",r)\n",
    "        return count\n",
    "    else: \n",
    "        #print(tuples)\n",
    "        #tuples.append(tuple(t))\n",
    "       \n",
    "        \n",
    "        print(\"Branch process\",t[0])\n",
    "        return count\n",
    "\n",
    "    \n",
    "def isClosed(t,g):\n",
    "    \"\"\"Checks whether g is closed in t.\"\"\"\n",
    "    cont=[]\n",
    "    for h in t:\n",
    "        if g&h==g:\n",
    "            cont.append(h)\n",
    "    if len(cont)>0:\n",
    "        res=cont[0]\n",
    "        for h in cont:\n",
    "            res=h&res\n",
    "        return res==g\n",
    "    else:\n",
    "        return g==0\n",
    "\n",
    "    \n",
    "def Check(t,nn1):\n",
    "    \"\"\"Checks whether T1 is fulfilled for t.\"\"\"\n",
    "    n=int(log(nn1+1,2))\n",
    "    members=[0]\n",
    "    members.extend([2**i for i in range(n)])\n",
    "    for m in members:\n",
    "        if not isClosed(t,m):\n",
    "            return False\n",
    "    return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def merge(r):\n",
    "    \"\"\"This is a value collector for the resulting list of dictionaries\n",
    "    \"\"\"\n",
    "    merged={}\n",
    "    for dic in r:\n",
    "        for key in dic:\n",
    "            if key in merged:\n",
    "                merged[key]+=dic[key]\n",
    "            else:\n",
    "                merged[key]=dic[key]\n",
    "    return merged"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Counting A334254 for n=6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
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0}, {(2, 59): 0, (3, 59): 0}, {(2, 59): 0}, {(2, 60): 0}, {(2, 60): 0}, {(2, 61): 0}]\n",
      "66960965307\n",
      "CPU times: user 9min 14s, sys: 8min 19s, total: 17min 33s\n",
      "Wall time: 4d 6h 37min 41s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "\n",
    "\n",
    "\n",
    "from multiprocess import Pool\n",
    "from multiprocess import cpu_count\n",
    "\n",
    "\n",
    "n=6\n",
    "\n",
    "#computing A334254 for n=6 by levels\n",
    "if __name__ == \"__main__\":\n",
    "    pool = Pool(cpu_count())\n",
    "    \n",
    "    nn1=2**n-1\n",
    "\n",
    "    lt=[]\n",
    "    lk=[]\n",
    "\n",
    "    count={}\n",
    "\n",
    "    \n",
    "    for t in range(nn1):\n",
    " \n",
    "        for i in range(t+1,nn1):\n",
    "            lt.append([t])\n",
    "            lk.append(i)\n",
    "            \n",
    "    ln_2=[nn1-1]* len(lt)\n",
    "    lnn1=[nn1]* len(lt)\n",
    "           \n",
    "   \n",
    "     \n",
    "    print(list(zip(lt,lk,ln_2,lnn1)))\n",
    "\n",
    "    \n",
    "    #parallel execution of ProcessRT function\n",
    "    res6 = pool.starmap_async(ProcessRT, zip(lt,lk,ln_2,lnn1))\n",
    "    print(res6.get())# print the list of resulting dictionaries\n",
    "    print(sum(merge(res6.get()).values()))#print the number of closure systems w.r.t. T1\n",
    "    pool.close()  # 'TERM'\n",
    "    pool.join()   # 'KILL'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "#saving the result to A334254_6.pickle file with pickle\n",
    "import pickle\n",
    "pikd = open(\"A334254_6.pickle\", \"wb\")\n",
    "pickle.dump(res6.get(), pikd)\n",
    "pikd.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
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       " {(2, 18): 0,\n",
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       " {(2, 18): 0,\n",
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       " {(2, 18): 0,\n",
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       "  (6, 18): 0,\n",
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       "  (14, 18): 0},\n",
       " {(2, 18): 0,\n",
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       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
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       "  (14, 18): 0},\n",
       " {(2, 18): 0,\n",
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       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0,\n",
       "  (9, 18): 0,\n",
       "  (10, 18): 0,\n",
       "  (11, 18): 0,\n",
       "  (12, 18): 0,\n",
       "  (13, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0,\n",
       "  (9, 18): 0,\n",
       "  (10, 18): 0,\n",
       "  (11, 18): 0},\n",
       " {(2, 18): 0,\n",
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       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0,\n",
       "  (9, 18): 0,\n",
       "  (10, 18): 0,\n",
       "  (11, 18): 0,\n",
       "  (12, 18): 0,\n",
       "  (13, 18): 0},\n",
       " {(2, 18): 0,\n",
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       "  (5, 18): 0,\n",
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       "  (10, 18): 0,\n",
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       "  (13, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
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       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
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       "  (9, 18): 0,\n",
       "  (10, 18): 0,\n",
       "  (11, 18): 0,\n",
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       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
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       "  (7, 18): 0,\n",
       "  (8, 18): 0,\n",
       "  (9, 18): 0,\n",
       "  (10, 18): 0,\n",
       "  (11, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0,\n",
       "  (9, 18): 0,\n",
       "  (10, 18): 0,\n",
       "  (11, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0,\n",
       "  (9, 18): 0,\n",
       "  (10, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0,\n",
       "  (9, 18): 0},\n",
       " {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0, (6, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0},\n",
       " {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0, (6, 18): 0, (7, 18): 0},\n",
       " {(2, 18): 0,\n",
       "  (3, 18): 0,\n",
       "  (4, 18): 0,\n",
       "  (5, 18): 0,\n",
       "  (6, 18): 0,\n",
       "  (7, 18): 0,\n",
       "  (8, 18): 0},\n",
       " {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0, (6, 18): 0, (7, 18): 0},\n",
       " {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0, (6, 18): 0},\n",
       " ...]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res6.get()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "66960965307"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(merge(res6.get()).values())# the computed value A334254 for n=6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "scrolled": true
   },
   "source": [
    "# 3. Checking A334254 for n=3, 4, 5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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543, (9, 13): 150, (10, 13): 11}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 3, (6, 13): 82, (7, 13): 264, (8, 13): 190, (9, 13): 21}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0, (6, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0, (6, 13): 0, (7, 13): 0, (8, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0, (6, 13): 0, (7, 13): 0, (8, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0, (6, 13): 0, (7, 13): 0, (8, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0, (6, 13): 0, (7, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0, (6, 13): 0, (7, 13): 0, (8, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0, (6, 13): 0, (7, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0, (6, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0, (5, 13): 0}, {(2, 13): 0, (3, 13): 0, (4, 13): 0}, {(2, 13): 0, (3, 13): 0}, {(2, 13): 0, (3, 13): 0}, {(2, 13): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 3, (6, 14): 82, (7, 14): 264, (8, 14): 190, (9, 14): 21}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0, (6, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0, (6, 14): 0, (7, 14): 0, (8, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0, (6, 14): 0, (7, 14): 0, (8, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0, (6, 14): 0, (7, 14): 0, (8, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0, (6, 14): 0, (7, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0, (6, 14): 0, (7, 14): 0, (8, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0, (6, 14): 0, (7, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0, (6, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0, (5, 14): 0}, {(2, 14): 0, (3, 14): 0, (4, 14): 0}, {(2, 14): 0, (3, 14): 0}, {(2, 14): 0, (3, 14): 0}, {(2, 14): 0}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 0, (6, 15): 0}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 5, (6, 15): 62, (7, 15): 117, (8, 15): 18}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 4, (6, 15): 37, (7, 15): 50, (8, 15): 6}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 10, (6, 15): 76, (7, 15): 79, (8, 15): 9}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 1, (6, 15): 7, (7, 15): 4}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 3, (6, 15): 24, (7, 15): 15, (8, 15): 1}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 0, (6, 15): 6, (7, 15): 3}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 2, (6, 15): 6}, {(2, 15): 0, (3, 15): 0, (4, 15): 0}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 0}, {(2, 15): 0, (3, 15): 0, (4, 15): 0, (5, 15): 0}, {(2, 15): 0, (3, 15): 0, (4, 15): 0}, {(2, 15): 0, (3, 15): 0}, {(2, 15): 0, (3, 15): 0}, {(2, 15): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0, (5, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0, (5, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0, (5, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0, (5, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0, (5, 16): 0}, {(2, 16): 0, (3, 16): 0, (4, 16): 0}, {(2, 16): 0, (3, 16): 0}, {(2, 16): 0, (3, 16): 0}, {(2, 16): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0, (5, 17): 0, (6, 17): 0, (7, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0, (5, 17): 0, (6, 17): 0, (7, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0, (5, 17): 0, (6, 17): 0, (7, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0, (5, 17): 0, (6, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0, (5, 17): 0, (6, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0, (5, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0, (5, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0, (5, 17): 0}, {(2, 17): 0, (3, 17): 0, (4, 17): 0}, {(2, 17): 0, (3, 17): 0}, {(2, 17): 0, (3, 17): 0}, {(2, 17): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0, (6, 18): 0, (7, 18): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0, (6, 18): 0, (7, 18): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0, (6, 18): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0, (6, 18): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0, (5, 18): 0}, {(2, 18): 0, (3, 18): 0, (4, 18): 0}, {(2, 18): 0, (3, 18): 0}, {(2, 18): 0, (3, 18): 0}, {(2, 18): 0}, {(2, 19): 0, (3, 19): 0, (4, 19): 0, (5, 19): 0, (6, 19): 0}, {(2, 19): 0, (3, 19): 0, (4, 19): 0, (5, 19): 0, (6, 19): 0, (7, 19): 0}, {(2, 19): 0, (3, 19): 0, (4, 19): 0, (5, 19): 0, (6, 19): 0, (7, 19): 0}, {(2, 19): 0, (3, 19): 0, (4, 19): 0, (5, 19): 0, (6, 19): 0}, {(2, 19): 0, (3, 19): 0, (4, 19): 0}, {(2, 19): 0, (3, 19): 0, (4, 19): 0, (5, 19): 0}, {(2, 19): 0, (3, 19): 0, (4, 19): 0, (5, 19): 0}, {(2, 19): 0, (3, 19): 0, (4, 19): 0}, {(2, 19): 0, (3, 19): 0}, {(2, 19): 0, (3, 19): 0}, {(2, 19): 0}, {(2, 20): 0, (3, 20): 0, (4, 20): 0, (5, 20): 0, (6, 20): 0}, {(2, 20): 0, (3, 20): 0, (4, 20): 0, (5, 20): 0, (6, 20): 0}, {(2, 20): 0, (3, 20): 0, (4, 20): 0, (5, 20): 0}, {(2, 20): 0, (3, 20): 0, (4, 20): 0}, {(2, 20): 0, (3, 20): 0, (4, 20): 0, (5, 20): 0}, {(2, 20): 0, (3, 20): 0, (4, 20): 0, (5, 20): 0}, {(2, 20): 0, (3, 20): 0, (4, 20): 0}, {(2, 20): 0, (3, 20): 0}, {(2, 20): 0, (3, 20): 0}, {(2, 20): 0}, {(2, 21): 0, (3, 21): 0, (4, 21): 0, (5, 21): 0, (6, 21): 0, (7, 21): 0}, {(2, 21): 0, (3, 21): 0, (4, 21): 0, (5, 21): 0, (6, 21): 0}, {(2, 21): 0, (3, 21): 0, (4, 21): 0}, {(2, 21): 0, (3, 21): 0, (4, 21): 0, (5, 21): 0}, {(2, 21): 0, (3, 21): 0, (4, 21): 0, (5, 21): 0}, {(2, 21): 0, (3, 21): 0, (4, 21): 0}, {(2, 21): 0, (3, 21): 0}, {(2, 21): 0, (3, 21): 0}, {(2, 21): 0}, {(2, 22): 0, (3, 22): 0, (4, 22): 0, (5, 22): 0, (6, 22): 0}, {(2, 22): 0, (3, 22): 0, (4, 22): 0}, {(2, 22): 0, (3, 22): 0, (4, 22): 0, (5, 22): 0}, {(2, 22): 0, (3, 22): 0, (4, 22): 0, (5, 22): 0}, {(2, 22): 0, (3, 22): 0, (4, 22): 0}, {(2, 22): 0, (3, 22): 0}, {(2, 22): 0, (3, 22): 0}, {(2, 22): 0}, {(2, 23): 0, (3, 23): 0, (4, 23): 0}, {(2, 23): 0, (3, 23): 0, (4, 23): 0, (5, 23): 0}, {(2, 23): 0, (3, 23): 0, (4, 23): 0, (5, 23): 0}, {(2, 23): 0, (3, 23): 0, (4, 23): 0}, {(2, 23): 0, (3, 23): 0}, {(2, 23): 0, (3, 23): 0}, {(2, 23): 0}, {(2, 24): 0, (3, 24): 0}, {(2, 24): 0, (3, 24): 0}, {(2, 24): 0, (3, 24): 0}, {(2, 24): 0, (3, 24): 0}, {(2, 24): 0, (3, 24): 0}, {(2, 24): 0}, {(2, 25): 0, (3, 25): 0, (4, 25): 0}, {(2, 25): 0, (3, 25): 0, (4, 25): 0}, {(2, 25): 0, (3, 25): 0}, {(2, 25): 0, (3, 25): 0}, {(2, 25): 0}, {(2, 26): 0, (3, 26): 0, (4, 26): 0}, {(2, 26): 0, (3, 26): 0}, {(2, 26): 0, (3, 26): 0}, {(2, 26): 0}, {(2, 27): 0, (3, 27): 0}, {(2, 27): 0, (3, 27): 0}, {(2, 27): 0}, {(2, 28): 0}, {(2, 28): 0}, {(2, 29): 0}]\n",
      "702525\n",
      "CPU times: user 59.3 ms, sys: 42.8 ms, total: 102 ms\n",
      "Wall time: 4.26 s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "\n",
    "\n",
    "\n",
    "from multiprocess import Pool\n",
    "from multiprocess import cpu_count\n",
    "\n",
    "\n",
    "n=5\n",
    "\n",
    "#computing A334254 for n=5 by levels\n",
    "if __name__ == \"__main__\":\n",
    "    pool = Pool(cpu_count())\n",
    "    \n",
    "    nn1=2**n-1\n",
    "\n",
    "    lt=[]\n",
    "    lk=[]\n",
    "\n",
    "    count={}\n",
    "\n",
    "    \n",
    "    for t in range(nn1):\n",
    " \n",
    "        for i in range(t+1,nn1):\n",
    "            lt.append([t])\n",
    "            lk.append(i)\n",
    "            \n",
    "    ln_2=[nn1-1]* len(lt)\n",
    "    lnn1=[nn1]* len(lt)\n",
    "           \n",
    "  \n",
    "     \n",
    "    print(list(zip(lt,lk,ln_2,lnn1)))\n",
    "\n",
    "    \n",
    "    #parallel execution of ProcessRT function\n",
    "    res5 = pool.starmap_async(ProcessRT, zip(lt,lk,ln_2,lnn1))\n",
    "    print(res5.get())# print the list of resulting dictionaries\n",
    "    print(sum(merge(res5.get()).values()))#print the number of closure systems w.r.t. T1\n",
    "\n",
    "    pool.close()  # 'TERM'\n",
    "    pool.join()   # 'KILL'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "702525"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(merge(res5.get()).values())# the computed value A334254 for n=5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[([0], 1, 14, 15), ([0], 2, 14, 15), ([0], 3, 14, 15), ([0], 4, 14, 15), ([0], 5, 14, 15), ([0], 6, 14, 15), ([0], 7, 14, 15), ([0], 8, 14, 15), ([0], 9, 14, 15), ([0], 10, 14, 15), ([0], 11, 14, 15), ([0], 12, 14, 15), ([0], 13, 14, 15), ([0], 14, 14, 15), ([1], 2, 14, 15), ([1], 3, 14, 15), ([1], 4, 14, 15), ([1], 5, 14, 15), ([1], 6, 14, 15), ([1], 7, 14, 15), ([1], 8, 14, 15), ([1], 9, 14, 15), ([1], 10, 14, 15), ([1], 11, 14, 15), ([1], 12, 14, 15), ([1], 13, 14, 15), ([1], 14, 14, 15), ([2], 3, 14, 15), ([2], 4, 14, 15), ([2], 5, 14, 15), ([2], 6, 14, 15), ([2], 7, 14, 15), ([2], 8, 14, 15), ([2], 9, 14, 15), ([2], 10, 14, 15), ([2], 11, 14, 15), ([2], 12, 14, 15), ([2], 13, 14, 15), ([2], 14, 14, 15), ([3], 4, 14, 15), ([3], 5, 14, 15), ([3], 6, 14, 15), ([3], 7, 14, 15), ([3], 8, 14, 15), ([3], 9, 14, 15), ([3], 10, 14, 15), ([3], 11, 14, 15), ([3], 12, 14, 15), ([3], 13, 14, 15), ([3], 14, 14, 15), ([4], 5, 14, 15), ([4], 6, 14, 15), ([4], 7, 14, 15), ([4], 8, 14, 15), ([4], 9, 14, 15), ([4], 10, 14, 15), ([4], 11, 14, 15), ([4], 12, 14, 15), ([4], 13, 14, 15), ([4], 14, 14, 15), ([5], 6, 14, 15), ([5], 7, 14, 15), ([5], 8, 14, 15), ([5], 9, 14, 15), ([5], 10, 14, 15), ([5], 11, 14, 15), ([5], 12, 14, 15), ([5], 13, 14, 15), ([5], 14, 14, 15), ([6], 7, 14, 15), ([6], 8, 14, 15), ([6], 9, 14, 15), ([6], 10, 14, 15), ([6], 11, 14, 15), ([6], 12, 14, 15), ([6], 13, 14, 15), ([6], 14, 14, 15), ([7], 8, 14, 15), ([7], 9, 14, 15), ([7], 10, 14, 15), ([7], 11, 14, 15), ([7], 12, 14, 15), ([7], 13, 14, 15), ([7], 14, 14, 15), ([8], 9, 14, 15), ([8], 10, 14, 15), ([8], 11, 14, 15), ([8], 12, 14, 15), ([8], 13, 14, 15), ([8], 14, 14, 15), ([9], 10, 14, 15), ([9], 11, 14, 15), ([9], 12, 14, 15), ([9], 13, 14, 15), ([9], 14, 14, 15), ([10], 11, 14, 15), ([10], 12, 14, 15), ([10], 13, 14, 15), ([10], 14, 14, 15), ([11], 12, 14, 15), ([11], 13, 14, 15), ([11], 14, 14, 15), ([12], 13, 14, 15), ([12], 14, 14, 15), ([13], 14, 14, 15)]\n",
      "[{(2, 0): 0, (3, 0): 0, (4, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0, (5, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0, (5, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0, (5, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0, (5, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0, (5, 0): 0}, {(2, 0): 0, (3, 0): 0, (4, 0): 0}, {(2, 0): 0, (3, 0): 0}, {(2, 0): 0, (3, 0): 0}, {(2, 0): 0}, {(2, 1): 0, (3, 1): 0, (4, 1): 1, (5, 1): 25, (6, 1): 67, (7, 1): 12}, {(2, 1): 0, (3, 1): 0, (4, 1): 0, (5, 1): 8, (6, 1): 21, (7, 1): 5}, {(2, 1): 0, (3, 1): 0, (4, 1): 0, (5, 1): 5, (6, 1): 13, (7, 1): 1}, {(2, 1): 0, (3, 1): 0, (4, 1): 0, (5, 1): 4, (6, 1): 8}, {(2, 1): 0, (3, 1): 0, (4, 1): 3, (5, 1): 15, (6, 1): 8}, {(2, 1): 0, (3, 1): 0, (4, 1): 1, (5, 1): 5}, {(2, 1): 0, (3, 1): 0, (4, 1): 0}, {(2, 1): 0, (3, 1): 0, (4, 1): 0, (5, 1): 0}, {(2, 1): 0, (3, 1): 0, (4, 1): 0, (5, 1): 0}, {(2, 1): 0, (3, 1): 0, (4, 1): 0}, {(2, 1): 0, (3, 1): 0}, {(2, 1): 0, (3, 1): 0}, {(2, 1): 0}, {(2, 2): 0, (3, 2): 0, (4, 2): 0, (5, 2): 8, (6, 2): 21, (7, 2): 5}, {(2, 2): 0, (3, 2): 0, (4, 2): 0, (5, 2): 5, (6, 2): 13, (7, 2): 1}, {(2, 2): 0, (3, 2): 0, (4, 2): 3, (5, 2): 18, (6, 2): 14}, {(2, 2): 0, (3, 2): 0, (4, 2): 0, (5, 2): 1, (6, 2): 2}, {(2, 2): 0, (3, 2): 0, (4, 2): 1, (5, 2): 5}, {(2, 2): 0, (3, 2): 0, (4, 2): 0}, {(2, 2): 0, (3, 2): 0, (4, 2): 0, (5, 2): 0}, {(2, 2): 0, (3, 2): 0, (4, 2): 0, (5, 2): 0}, {(2, 2): 0, (3, 2): 0, (4, 2): 0}, {(2, 2): 0, (3, 2): 0}, {(2, 2): 0, (3, 2): 0}, {(2, 2): 0}, {(2, 3): 0, (3, 3): 0, (4, 3): 3, (5, 3): 21, (6, 3): 20}, {(2, 3): 0, (3, 3): 0, (4, 3): 4, (5, 3): 31, (6, 3): 44, (7, 3): 8}, {(2, 3): 0, (3, 3): 0, (4, 3): 3, (5, 3): 18, (6, 3): 12, (7, 3): 1}, {(2, 3): 0, (3, 3): 0, (4, 3): 0, (5, 3): 6, (6, 3): 3}, {(2, 3): 0, (3, 3): 0, (4, 3): 0}, {(2, 3): 0, (3, 3): 0, (4, 3): 0, (5, 3): 0}, {(2, 3): 0, (3, 3): 0, (4, 3): 0, (5, 3): 0}, {(2, 3): 0, (3, 3): 0, (4, 3): 0}, {(2, 3): 0, (3, 3): 0}, {(2, 3): 0, (3, 3): 0}, {(2, 3): 0}, {(2, 4): 0, (3, 4): 0, (4, 4): 0, (5, 4): 1, (6, 4): 2}, {(2, 4): 0, (3, 4): 0, (4, 4): 0, (5, 4): 1, (6, 4): 2}, {(2, 4): 0, (3, 4): 0, (4, 4): 1, (5, 4): 5}, {(2, 4): 0, (3, 4): 0, (4, 4): 0}, {(2, 4): 0, (3, 4): 0, (4, 4): 0, (5, 4): 0}, {(2, 4): 0, (3, 4): 0, (4, 4): 0, (5, 4): 0}, {(2, 4): 0, (3, 4): 0, (4, 4): 0}, {(2, 4): 0, (3, 4): 0}, {(2, 4): 0, (3, 4): 0}, {(2, 4): 0}, {(2, 5): 0, (3, 5): 0, (4, 5): 3, (5, 5): 18, (6, 5): 12, (7, 5): 1}, {(2, 5): 0, (3, 5): 0, (4, 5): 0, (5, 5): 6, (6, 5): 3}, {(2, 5): 0, (3, 5): 0, (4, 5): 0}, {(2, 5): 0, (3, 5): 0, (4, 5): 0, (5, 5): 0}, {(2, 5): 0, (3, 5): 0, (4, 5): 0, (5, 5): 0}, {(2, 5): 0, (3, 5): 0, (4, 5): 0}, {(2, 5): 0, (3, 5): 0}, {(2, 5): 0, (3, 5): 0}, {(2, 5): 0}, {(2, 6): 0, (3, 6): 0, (4, 6): 0, (5, 6): 6, (6, 6): 3}, {(2, 6): 0, (3, 6): 0, (4, 6): 0}, {(2, 6): 0, (3, 6): 0, (4, 6): 0, (5, 6): 0}, {(2, 6): 0, (3, 6): 0, (4, 6): 0, (5, 6): 0}, {(2, 6): 0, (3, 6): 0, (4, 6): 0}, {(2, 6): 0, (3, 6): 0}, {(2, 6): 0, (3, 6): 0}, {(2, 6): 0}, {(2, 7): 0, (3, 7): 0, (4, 7): 0}, {(2, 7): 0, (3, 7): 0, (4, 7): 1, (5, 7): 5}, {(2, 7): 0, (3, 7): 0, (4, 7): 0, (5, 7): 1}, {(2, 7): 0, (3, 7): 0, (4, 7): 1}, {(2, 7): 0, (3, 7): 0}, {(2, 7): 0, (3, 7): 0}, {(2, 7): 0}, {(2, 8): 0, (3, 8): 0}, {(2, 8): 0, (3, 8): 0}, {(2, 8): 0, (3, 8): 0}, {(2, 8): 0, (3, 8): 0}, {(2, 8): 0, (3, 8): 0}, {(2, 8): 0}, {(2, 9): 0, (3, 9): 0, (4, 9): 0}, {(2, 9): 0, (3, 9): 0, (4, 9): 0}, {(2, 9): 0, (3, 9): 0}, {(2, 9): 0, (3, 9): 0}, {(2, 9): 0}, {(2, 10): 0, (3, 10): 0, (4, 10): 0}, {(2, 10): 0, (3, 10): 0}, {(2, 10): 0, (3, 10): 0}, {(2, 10): 0}, {(2, 11): 0, (3, 11): 0}, {(2, 11): 0, (3, 11): 0}, {(2, 11): 0}, {(2, 12): 0}, {(2, 12): 0}, {(2, 13): 0}]\n",
      "545\n",
      "CPU times: user 31 ms, sys: 43.9 ms, total: 75 ms\n",
      "Wall time: 86 ms\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "\n",
    "\n",
    "\n",
    "from multiprocess import Pool\n",
    "from multiprocess import cpu_count\n",
    "\n",
    "\n",
    "n=4\n",
    "\n",
    "#computing A334254 for n=4 by levels\n",
    "if __name__ == \"__main__\":\n",
    "    pool = Pool(cpu_count())\n",
    "    \n",
    "    nn1=2**n-1\n",
    "\n",
    "    lt=[]\n",
    "    lk=[]\n",
    "\n",
    "    count={}\n",
    "\n",
    "    \n",
    "    for t in range(nn1):\n",
    " \n",
    "        for i in range(t+1,nn1):\n",
    "            lt.append([t])\n",
    "            lk.append(i)\n",
    "            \n",
    "    ln_2=[nn1-1]* len(lt)\n",
    "    lnn1=[nn1]* len(lt)\n",
    "           \n",
    "  \n",
    "     \n",
    "    print(list(zip(lt,lk,ln_2,lnn1)))\n",
    "\n",
    "    \n",
    "    #parallel execution of ProcessRT function\n",
    "    res4 = pool.starmap_async(ProcessRT, zip(lt,lk,ln_2,lnn1))\n",
    "    print(res4.get())# print the list of resulting dictionaries\n",
    "    print(sum(merge(res4.get()).values()))#print the number of closure systems w.r.t. T1\n",
    "\n",
    "    pool.close()  # 'TERM'\n",
    "    pool.join()   # 'KILL'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "545"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(merge(res4.get()).values())# the computed value A334254 for n=4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[([0], 1, 6, 7), ([0], 2, 6, 7), ([0], 3, 6, 7), ([0], 4, 6, 7), ([0], 5, 6, 7), ([0], 6, 6, 7), ([1], 2, 6, 7), ([1], 3, 6, 7), ([1], 4, 6, 7), ([1], 5, 6, 7), ([1], 6, 6, 7), ([2], 3, 6, 7), ([2], 4, 6, 7), ([2], 5, 6, 7), ([2], 6, 6, 7), ([3], 4, 6, 7), ([3], 5, 6, 7), ([3], 6, 6, 7), ([4], 5, 6, 7), ([4], 6, 6, 7), ([5], 6, 6, 7)]\n",
      "[{(2, 0): 0, (3, 0): 0}, {(2, 0): 0, (3, 0): 0}, {(2, 0): 0, (3, 0): 0}, {(2, 0): 0, (3, 0): 0}, {(2, 0): 0, (3, 0): 0}, {(2, 0): 0}, {(2, 1): 0, (3, 1): 1, (4, 1): 4}, {(2, 1): 0, (3, 1): 0, (4, 1): 1}, {(2, 1): 0, (3, 1): 0}, {(2, 1): 0, (3, 1): 0}, {(2, 1): 0}, {(2, 2): 0, (3, 2): 0, (4, 2): 1}, {(2, 2): 0, (3, 2): 0}, {(2, 2): 0, (3, 2): 0}, {(2, 2): 0}, {(2, 3): 0, (3, 3): 0}, {(2, 3): 0, (3, 3): 1}, {(2, 3): 0}, {(2, 4): 0}, {(2, 4): 0}, {(2, 5): 0}]\n",
      "8\n",
      "CPU times: user 21 ms, sys: 32 ms, total: 53.1 ms\n",
      "Wall time: 57 ms\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "\n",
    "\n",
    "\n",
    "from multiprocess import Pool\n",
    "from multiprocess import cpu_count\n",
    "\n",
    "\n",
    "n=3\n",
    "\n",
    "#computing A334254 for n=3 by levels\n",
    "if __name__ == \"__main__\":\n",
    "    pool = Pool(cpu_count())\n",
    "    \n",
    "    nn1=2**n-1\n",
    "\n",
    "    lt=[]\n",
    "    lk=[]\n",
    "\n",
    "    count={}\n",
    "\n",
    "    \n",
    "    for t in range(nn1):\n",
    " \n",
    "        for i in range(t+1,nn1):\n",
    "            lt.append([t])\n",
    "            lk.append(i)\n",
    "            \n",
    "    ln_2=[nn1-1]* len(lt)\n",
    "    lnn1=[nn1]* len(lt)\n",
    "           \n",
    "  \n",
    "     \n",
    "    print(list(zip(lt,lk,ln_2,lnn1)))\n",
    "\n",
    "    \n",
    "    #parallel execution of ProcessRT function\n",
    "    res3 = pool.starmap_async(ProcessRT, zip(lt,lk,ln_2,lnn1))\n",
    "    print(res3.get())# print the list of resulting dictionaries\n",
    "    print(sum(merge(res3.get()).values()))#print the number of closure systems w.r.t. T1\n",
    "\n",
    "    pool.close()  # 'TERM'\n",
    "    pool.join()   # 'KILL'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(merge(res3.get()).values())# the computed value A334254 for n=3"
   ]
  }
 ],
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