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# tuttePolynomial(Matroid) -- Tutte polynomial of a matroid

- Function: tuttePolynomial
- Usage:
    `tuttePolynomial M`
- Inputs:
  - M, a matroid,
- Outputs:
  - a ring element, the Tutte polynomial of M

## Description

The Tutte polynomial is an invariant of a matroid that is universal with respect to satisfying a deletion-contraction recurrence. Indeed, one way to define the Tutte polynomial of a matroid is: if $M$ is a matroid consisting of $a$ loops and $b$ coloops, then $T_M(x, y) = x^ay^b$, and if $e \in M$ is neither a loop nor a coloop, then $T_M(x, y) := T_{M \backslash e}(x, y) + T_{M/e}(x, y)$, where M\e is the deletion of M with respect to $\{e\}$, and M/e is the contraction of M with respect to $\{e\}$. Many invariants of a matroid can be determined by substituting values into its Tutte polynomial - cf. tutteEvaluate.

    i1 : tuttePolynomial matroid completeGraph 4

          3    3     2            2
    o1 = x  + y  + 3x  + 4x*y + 3y  + 2x + 2y

    o1 : ZZ[x..y]

    i2 : tuttePolynomial specificMatroid "nonpappus"

          6     5     4    3      3     2             2
    o2 = y  + 3y  + 6y  + x  + 10y  + 6x  + 8x*y + 15y  + 13x + 13y

    o2 : ZZ[x..y]

## See also

- tutteEvaluate -- evaluate Tutte polynomial
- characteristicPolynomial(Matroid) -- computes characteristic polynomial of a matroid
- chromaticPolynomial -- computes chromatic polynomial of a graph
- deletion -- deletion of subset of matroid
- contraction -- contraction of subset of matroid

## Ways to use this method:

- tuttePolynomial(Matroid) -- Tutte polynomial of a matroid
- `tuttePolynomial(Matroid,Ring)`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Matroids/doc-Matroids.m2:2556:0.