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# rookPolynomial -- Rook polynomial of a collection of cells

- Usage:
    `rookPolynomial Q`
- Inputs:
  - Q, a collection of cells
- Outputs:
  - the rook polynomial of Q

## Description

The rook polynomial of a collection of cells $\mathcal{P}$ is defined as $r_\mathcal{P}(t) = \sum_{k \ge 0} r_k(\mathcal{P}) t^k,$ where $r_k(\mathcal{P})$ denotes the number of non-attacking configurations of $k$ rooks on $\mathcal{P}$. This function computes and returns the polynomial $r_\mathcal{P}(t)$.
The importance of this rook polynomial is underlined by the fact that it appears to describe the Hilbert series of the coordinate ring of collections of cells that do not contain the square tetromino, also known as thin (see [RR2021]).

- [RR2021] Hilbert series of simple thin polyominoes(R. Rinaldo, F. Romeo, 2021, J. Algebr. Comb.).

    i1 : Q = cellCollection {{1,1},{1,2},{2,1},{2,2}};

    i2 : rookPolynomial Q

           2
    o2 = 2t  + 4t + 1

    o2 : ZZ[t]

    i3 : Q = cellCollection {{1,1},{1,2},{2,1},{3,1},{3,2}};

    i4 : rookPolynomial Q

          3     2
    o4 = t  + 5t  + 5t + 1

    o4 : ZZ[t]

## See also

- cellCollection -- Create a collection of cells
- isNonAttackingRooks -- Check if a rook configuration is non-attacking
- allNonAttackingRookConfigurations -- All non-attacking rook configurations in Q
- rookNumber -- Rook number of a collection of cells

## Ways to use rookPolynomial:

- `rookPolynomial(CollectionOfCells)`

## For the programmer

The object rookPolynomial is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PolyominoIdeals.m2:1400:0.