# tuttePolynomial -- computes the Tutte polynomial of a poset

## Synopsis

• Usage:
f = tuttePolynomial P
• Inputs:
• P, an instance of the type Poset,
• Outputs:
• f, , 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