{
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   "id": "53aab533-e7f5-4cb4-a36f-a0f790749bf8",
   "metadata": {},
   "outputs": [],
   "source": [
    "import os \n",
    "import sys\n",
    "sys.path.insert(0, os.path.abspath('../..'))\n",
    "\n",
    "import sympy as sp\n",
    "import numpy as np\n",
    "import itertools as it\n",
    "import pysymmpol as sy\n",
    "import pysymmpol.utils as ut\n",
    "from IPython.display import display, Latex"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9b4fe6d-94db-48a6-927e-d2be77cc898d",
   "metadata": {},
   "source": [
    "# Class: MonomialPolynomial"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a6512c7-6b3d-45cb-b673-9e7582de69aa",
   "metadata": {},
   "source": [
    "Now we turn our attention to the Monomial Symmetric Polynomials. In terms of implementation, they are defined via inheritance of the Hall-Littlewood defined below, but since they are more fundamental than the Hall-Littlewood polynomials, let us treat them here.\n",
    "\n",
    "In terms of construction, we follow the logic of the previous classes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d6c48364-ee22-431c-b47a-59f8bac1aded",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We first define the partition and the Young diagram\n",
    "part = (3,2)\n",
    "yg = sy.YoungDiagram(part)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b2c854c3-e149-4076-bc79-57c12051ebac",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Then we call the class from the Young Diagram\n",
    "mp = sy.MonomialPolynomial(yg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "abba6a0b-d224-468e-a52b-0b1b4cacfcf6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x1, x2)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Next we build the x coordinates\n",
    "n = 2\n",
    "x = ut.create_x_coord(n)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f92cc030-0994-4ebc-bdc5-3b86a7831256",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x_{1}^{3} x_{2}^{2} + x_{1}^{2} x_{2}^{3}$"
      ],
      "text/plain": [
       "x1**3*x2**2 + x1**2*x2**3"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The explicit expression is (factor and expand are sympy methods\n",
    "mp.explicit(x).factor().expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "c67a5d31-6f10-4d99-8f8a-c404b95af8a8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\operatorname{Poly}{\\left( x_{1}^{3}x_{2}^{2} + x_{1}^{2}x_{2}^{3}, x_{1}, x_{2}, domain=\\mathbb{Q} \\right)}$"
      ],
      "text/plain": [
       "Poly(x1**3*x2**2 + x1**2*x2**3, x1, x2, domain='QQ')"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The explicit expression is (factor and expand are sympy methods\n",
    "mp.explicit(x, True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "cafc14d5-9aa6-4765-9d12-1314652f7728",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x1,)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Next we build the x coordinates with n < len(part)\n",
    "n = 1\n",
    "x = ut.create_x_coord(n)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "a495a5be-01a4-417b-aea4-9b0dab425725",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mp.explicit(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "93a81b8f-5461-4cb0-b6fc-951aa1d39f30",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x1, x2, x3)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Next we build the x coordinates with n > len(part)\n",
    "n = 3\n",
    "x = ut.create_x_coord(n)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "e96b2b62-d17d-41f1-a4b7-bda08b6833d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x_{1}^{3} x_{2}^{2} + x_{1}^{3} x_{3}^{2} + x_{1}^{2} x_{2}^{3} + x_{1}^{2} x_{3}^{3} + x_{2}^{3} x_{3}^{2} + x_{2}^{2} x_{3}^{3}$"
      ],
      "text/plain": [
       "x1**3*x2**2 + x1**3*x3**2 + x1**2*x2**3 + x1**2*x3**3 + x2**3*x3**2 + x2**2*x3**3"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mp.explicit(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a612b0c-63d0-4bd0-9e10-47404d430462",
   "metadata": {},
   "source": [
    "We also have a getter for the partition associated to a given Monomial Symmetric Polynomial."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "087a2f31-9d6c-46f2-a561-556e54cbf556",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 2)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mp.partition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a01f7a03-cceb-42fe-88c0-602f4c0767bd",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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