{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is an ipython (jupyter) worksheet with code from the lecture notes for the course\n",
    "**Permutation Puzzles: A Mathematical Perspective**, by Jamie Mulholland\n",
    "\n",
    "Coures webpage: http://www.sfu.ca/~jtmulhol/math302 <br />\n",
    "Course notes booklet: http://www.sfu.ca/~jtmulhol/math302/notes/302notes.pdf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 3: Permutations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Section 3.1 Permutation: Preliminary Definition\n",
    "\n",
    "Curly braces { } denote *sets*,  i.e. the order that elements are listed doesn't matter.  Square braces [ ] denote lists, i.e. the order that elements appear does matter.  So as sets {1,2,3} = {2,1,3} but as lists [1,2,3] $\\neq$ [2,1,3]."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Set([1,2,3])==Set([2,1,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[1,2,3]==[2,1,3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example 3.2**: Find the number of ways to arrange 7 books on a shelf. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5040"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial(7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use SageMath to generate permutations of a list, for example [1,2,3]."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Permutations of the set [1, 2, 3]\n"
     ]
    }
   ],
   "source": [
    "terms=[1,2,3]\n",
    "p = Permutations(terms)\n",
    "print p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.list()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.cardinality()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example 3.5:** Two permutations of the multi-set [a,a,b,b,b] are [b,a,b,a,b] and [b,b,a,a,b].  There are $\\dfrac{5!}{2!\\cdot3!}=10$ permutations in total."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Permutations of the multi-set [a, a, b, b, b]\n"
     ]
    }
   ],
   "source": [
    "var('a,b');\n",
    "terms=[a,a,b,b,b];\n",
    "p = Permutations(terms)\n",
    "print p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[a, a, b, b, b],\n",
       " [a, b, a, b, b],\n",
       " [a, b, b, a, b],\n",
       " [a, b, b, b, a],\n",
       " [b, a, a, b, b],\n",
       " [b, a, b, a, b],\n",
       " [b, a, b, b, a],\n",
       " [b, b, a, a, b],\n",
       " [b, b, a, b, a],\n",
       " [b, b, b, a, a]]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.list()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.cardinality()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial(5)/(factorial(2)*factorial(3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Section 3.2 Permutation: Mathematical Definition\n",
    "\n",
    "### Section 3.2.2 Permutations\n",
    "\n",
    "A **permutation** of a set A is a function $\\alpha : A \\rightarrow A$ that is bijective (i.e. both one-to-one and onto).\n",
    "\n",
    "For example, we can define a permutation $\\alpha$ of the set $\\{1,2,3\\}$ by stating:\n",
    "$$\\alpha(1)=2, \\quad \\alpha(2)=1, \\quad \\alpha(3)=3.$$ \n",
    "\n",
    "In SageMath we can use the `Permutation()` command to construct a permutation.  Here we define the permutation by the list of images $[\\alpha(1), \\alpha(2), \\ldots]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2, 1, 3]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a=Permutation([2,1,3]); a   # permutation which maps 1->2, 2->1, 3->3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is an example of how to use matrices in SageMath to display a permutation in array form.  We can use the `matrix()` command, where the syntax is the following. <br />\n",
    "`matrix( [ <list for row 1> , <list for row 2> ] )`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1 2 3]\n",
       "[2 1 3]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a=Permutation([2,1,3])\n",
    "matrix([[1,2,3],[a(i) for i in [1,2,3]]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Section 3.3 Composing Permutations\n",
    "\n",
    "Let $\\alpha$ and $\\beta$ be two permutations of [1,...,n]. We wish to define a new permutation $\\alpha\\circ \\beta$, called the *permutation composition*. In order to define a function we just need to specify how it maps the elements.  For $k\\in [n]$ we'll define $(\\alpha\\circ \\beta)(k)$ to be the result of first applying $\\alpha$, then applying $\\beta$ to the result.  In other words, \n",
    "\n",
    "$$(\\alpha \\circ \\beta) (k) = \\beta(\\alpha(k)), \\text{ for $k \\in [n]$.}$$ \n",
    "\n",
    "(Warning: Notice that the composition is from left-to-right, which is opposite to the way functions were combined in calculus.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a = [5, 3, 1, 4, 2]\n",
      "b = [5, 3, 2, 1, 4]\n"
     ]
    }
   ],
   "source": [
    "a=Permutation([5,3,1,4,2]);  print \"a =\", a\n",
    "b=Permutation([5,3,2,1,4]); print \"b =\", b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4, 2, 5, 1, 3]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a*b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Permutation composition is not commutative in general.  That is, we typically have $\\alpha\\beta \\ne\\beta\\alpha$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2, 1, 3, 5, 4]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b*a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Section 3.5 Inverses of Permutation\n",
    "\n",
    "SageMath has a built in `inverse` command."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2, 3, 1, 5, 4]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a=Permutation([3,1,2,5,4])\n",
    "a.inverse()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 4, 1, 2]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b=Permutation([3,4,1,2])\n",
    "b.inverse()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br /> <br />\n",
    "\n",
    "**Note**: This lecture provided a very basic introduction to permutations in SageMath. We will be interested in doing algebra with permutations so we will use a different way to work with permuations in SageMath.  This will be introduced in Lecture 4, after we introduce the *Symmetric Group*.\n",
    "\n",
    "<br /> <br />"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 7.3",
   "language": "",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}