Computing the Tutte Polynomial of a Matroid from its Lattice of Cyclic Flats

We show how the Tutte polynomial of a matroid $M$ can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats. The results imply that the Tutte polynomial of $M$ is already determined by the abstract lattice of its cyclic flats together with their cardinalities and ranks. They furthermore generalize a similiar statement for perfect matroid designs due to Mphako and help to understand families of matroids with identical Tutte polynomial as constructed by Ken Shoda.


Introduction
The Tutte polynomial is a central invariant in matroid theory. But passing over from a matroid M to its Tutte polynomial T(M ; x, y) generally means a big loss of information. This paper gives one explanation for this phenomenon by showing how little information about the cyclic flats of a matroid is really needed for the computation of its Tutte polynomial.
From now on let M be a matroid. A flat X in M is called cyclic if M |X contains no coloops. Section 2 will recapitulate some basic facts about cyclic flats and show how the Tutte polynomial can be expressed in terms of cloud and flock polynomials of cyclic flats as introduced by Plesken in [6]. Then Section 3 establishes some important identities for cloud and flock polynomials needed later on.
Z(M ), the set of cyclic flats of M , is a lattice w.r.t. inclusion (c.f. Figure  1). In Section 4 we introduce the configuration of M : the abstract lattice of its cyclic flats 1 together with their cardinalities and ranks. We then prove: Theorem 4.1. The Tutte polynomial of a matroid is determined by its configuration.
While M is determined by its cyclic flats and their ranks (c.f. [1]), it generally is far from being determined by its configuration (c.f. Figure 1); there are even superexponential families of matroids with identical configurations (c.f. [7]). So Theorem 4.1 explains one big part of the information lost when passing from M to its Tutte polynomial.
In Section 5 we incorporate symmetries in M to shrink down the information needed for its Tutte polynomial even more. Let G ≤ Aut(M ), P be the set of G-orbits of Z(M ) and {R B } B∈P a system of representatives. The condensed configuration of M corresponding to P consists of the cardinalities and ranks of the R B and the matrix 2 (A P (B, C)) B,C∈P where After discussing some examples, e.g. a condensed configuration for the Golay code matroid, we will prove: Section 6 shows how to obtain a condensed configuration of a perfect matroid design using only the cardinalities of flats of given rank. Together with Theorem 5.1 this yields a new proof for Mphako's results about the Tutte polynomial of perfect matroid designs in [4].

Background
We quickly recapitulate the most important facts about cyclic flats and the cloud/flock formula for the rank generating polynomial from [6], while assuming familiarity with the basics of matroid theory. From now on let M be a matroid without loops and coloops with rank funktion r M , closure operator cl M and ground set E(M ).

Cyclic flats
A flat X in M is called a cyclic flat if M |X, the restriction of M to X, contains no coloops. We denote the set of (cyclic) flats by L(M ), resp. Z(M ); both form a lattice w.r.t. inclusion.

Identities for cloud and flock polynomials
We state some useful identities for cloud and flock polynomials and show that the cloud and flock polynomial of the empty set and the ground set are already determined by the cloud and flock polynomials of all other cyclic flats. This is crucial for the recursive algorithms introduced later on. The rank generating polynomial S(M ; x, y) of M is per definitionem a sum over all subsets of E(M ) and it is easy to see that for n = |E(M )| and r = r M (M ) To make use of this identity we define the Z-linear maps 4 And furthermore a notation for the cloud and flock polynomials in the uniform matroid U r,n . For r < n: and for n = r = 0 : c 0,0 (x) := f 0,0 (y) = 1.
In this notation equation (3.1) becomes Applying these identities to the cloud/flock formula of the rank generating polynomial we obtain . Another crucial fact is that the cloud (flock) polynomial of a cyclic flat X only depends on M/X (M |X) as the following lemma states.

Configurations
We introduce the configuration of a matroid and show how to compute the rank generating polynomial only using the configuration. The configuration of a matroid describes how it is is build up from uniform matroids as the following example illustrates: Example 3. 1.) According to Example 1, a matroid without coloops is uniform on n points and of rank r (r < n) iff its configuration is The main motivation for introducing configurations is the following main theorem. Theorem 4.1 yields a better understanding in matroids with identical rank generating polynomials. In [7] superexponential familes of matroids with identical rank generating polynomials are constructed; as it turns out they all have -by construction -the same configuration.

Condensed configurations
We show how to incorporate symmetries in M to shrink down the information needed for the computation of the rank generating polynomial even more by introducing the condensed configuration.
But first, we generalize the notion of the set of orbits of cyclic flats.
From this we can for example read off that the cyclic flats of cardinality 12 and rank 11 are neither contained in nor contain any other non trivial cyclic flats in M . Using the axiom scheme to define a matroid by its cyclic flats introduced in [1], we can safely remove this orbit of cyclic flats from Z(M ) and obtain a new interesting matroid, which still has M 24 as automorphism group, but cannot be found "in nature" like M .
Again the definition of a condensed configuration is motivated by the following generalization of Theorem 4.1: Theorem 5.1. The rank generating polynomial can be computed by a condensed configuration.
Recall that in the proof of Theorem 4.1 we actually showed how to compute the cloud and flock polynomials in all minors M |X/Y for cyclic flats X ⊂ Y . Generally, this is not possible in the case of condensed configurations, for the simple reason that from a condensed configuration we cannot derive exactly which minors appear in M . But we will be able to compute average cloud and flock polynomials.
So let P be a condensation of M , {R B } B∈P a system of representatives. To prove Theorem 5.1, we show how to compute the rank generating polynomial of M only using the condensed configuration corresponding to P , i.e. the matrix (A P (B, C)) B,C∈P and the cardinalities and ranks of the R B , for B ∈ P .  Notice that this implies that the right hand sides are independent of the choice of R C ∈ C, since the left hand sides are. Proof By induction the rightmost sum is independent of the choice of R D , so The three sums range over {(X, Y )|Y ∈ Z(M ), X ∈ B with X Y R C } and can hence be rearranged to The statement for the flock polynomial follows analogously.
Those average cloud and flock polynomials suffice for the computation of the rank generating polynomial and analogously to the cloud/flock formula we obtain: The condensed configuration is constructed to contain all necessary information for the computation of the c(P, B, C; x) and f(P, B, C; y). This proves Theorem 5.1.

Condensed Configurations of Perfect Matroid Designs
M is called a perfect matroid design if all flats in M of given rank i have the same cardinality k i . In [4] Mphako showed that the rank generating polynomial of a perfect matroid design is already determined by the numbers k i . We show how to recover a condensed configuration of a perfect matroid design by the numbers k i as well. Combined with Theorem 5.1 this yields a new prove for Mphako's result.
Let for now M be a perfect matroid design of rank r, k i the cardinality of a flat of rank i and B i the set of flats of rank i. Notice that B i consists of cyclic flats iff k i > k i−1 + 1. By dualizing the first formula in the proof of Theorem 3.6 in [2] we obtain that for all i ≤ j and Y ∈ B j Hence P := {B i |k i > k i−1 + 1} is a condensation of Z(M ). The condensed configuration of M corresponding to P only depends on the numbers k i since the A P (B i , B j ) and the cardinalities and ranks of the cyclic flats are determined by the k i . Summarizing this yields a new proof for: Theorem 6.1 (Mphako [4]). The rank generating ( Tutte) polynomial of perfect matroid design only depends on the cardinalities and ranks of its flats.
Notice that we actually proved a stronger statement, since we can moreover compute the average cloud and flock polynomials c(P, B i , B j ; x) and f(P, B i , B j ; y) now. This yields a new method to prove the nonexistence of certain perfect matroid designs. Firstly the coefficients of all average cloud and flock polynomials have to be positive integers. Secondly cardinality and rank of all flats which have to appear in the matroid can be determined by the exponents of the average cloud polynomials and may not differ from the k i .