Calculation of the frequencies distribution for the corners of all paths going to the right or upwards from the bottom-left to the top-right of a regular lattice of size n x m.

corners-distribuition.json

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "About"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we calculate the frequencies distribution for the corners of all paths going to the right or upwards from the bottom-left to the top-right of a regular lattice of size *n x m*."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy\n",
      "import matplotlib.pyplot as plt\n",
      "import numpy as np"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fact = sympy.combinatorial.numbers.factorial\n",
      "nC = sympy.combinatorial.numbers.nC"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def totalPaths(n, m):\n",
      "    return nC(n + m, n)\n",
      "\n",
      "\n",
      "def pathsWithOddCorners(n, m, r):\n",
      "    return 2 * nC((n - 1), (r - 1) / 2) * nC((m - 1), (r - 1) / 2)\n",
      "\n",
      "\n",
      "def pathsWithEvenCorners(n, m, r):\n",
      "    return nC(n - 1, r / 2) * nC(m - 1, (r - 2) / 2) + nC(n - 1, (r - 2) / 2) * nC(m - 1, r / 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n, m = 500, 300\n",
      "tpaths = totalPaths(n, m)\n",
      "\n",
      "evenCorners = np.array(xrange(2, 2 * min(n, m)))\n",
      "oddCorners = np.array(xrange(1, 2 * min(n, m) + 1, 2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "totalEvenCorners = [pathsWithEvenCorners(n, m, c) for c in evenCorners]\n",
      "totalOddCorners = [pathsWithOddCorners(n, m, c) for c in oddCorners]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "evenDistribuition = [corner / float(tpaths) for corner in totalEvenCorners]\n",
      "oddDistribuition = [corner / float(tpaths) for corner in totalOddCorners]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(evenCorners, evenDistribuition, 'r-', oddCorners, oddDistribuition, 'b-')\n",
      "plt.title('Corners distribuition in a %s x %s lattice' % (n, m))\n",
      "plt.xlabel('Number of corners')\n",
      "plt.ylabel('Corners density')\n",
      "plt.legend(('Even corners', 'Odd corners'), loc=\"upper left\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "<matplotlib.legend.Legend at 0x65dc650>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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EMnToULV06dIar3/55Zc1Ypk4caJKTExUSlUmlPj4eLPtwcHB6v333zc9/3//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\n1dDQgOnTp2PevHnKPCIt5WvWrVs3REdHo6CgQDP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       "text": [
        "<matplotlib.figure.Figure at 0x639cdd0>"
       ]
      }
     ],
     "prompt_number": 31
    }
   ],
   "metadata": {}
  }
 ]
}