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# tuttePolynomial -- computes the Tutte polynomial of a poset

- Usage:
    `f = tuttePolynomial P`
- Inputs:
  - P, an instance of the type Poset,
- Outputs:
  - f, a ring element, the Tutte polynomial of $P$

## Description

The Tutte polynomial of $P$ is a polynomial $f$ in two variables obtained by a summation over antichains.

    i1 : B = booleanLattice 3;

    i2 : f = tuttePolynomial B

          8 7     8 6      8 5      8 4      8 3    7 4      8 2     7 3     6 4
    o2 = t z  + 7t z  + 21t z  + 35t z  + 35t z  + t z  + 21t z  + 4t z  + 3t z
         ------------------------------------------------------------------------
             8      7 2      6 3    8     7       6 2     5 3    7      6
         + 7t z + 6t z  + 12t z  + t  + 4t z + 18t z  + 3t z  + t  + 12t z +
         ------------------------------------------------------------------------
           5 2     4 3     6     5      4 2     5      4      4     3      3
         9t z  + 3t z  + 3t  + 9t z + 9t z  + 3t  + 10t z + 4t  + 3t z + 3t  +
         ------------------------------------------------------------------------
           2      2
         3t z + 3t  + t + 1

    o2 : QQ[t, z]

The Tutte polynomial evaluated at $t = 1$ and $z = 1$ is always the number of subsets of the groundset of $P$.

    i3 : R = ring f;

    i4 : sub(f, {R_0 => 1, R_1 => 1})

    o4 = 256

    o4 : QQ

## See also

- antichains -- computes all antichains of a poset

## Ways to use tuttePolynomial:

- `tuttePolynomial(Poset)`

## For the programmer

The object tuttePolynomial is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Posets.m2:5323:0.