We establish a general theorem for reducing sums of type ∑ y≥x g(y) where g is a mapping from a partially ordered set into an abelian group. Conclusions concern the Möbius function, the principle of inclusionexclusion, the Tutte polynomial and Crapo’s beta invariant.

We formulate and prove a formula to compute the expected value of the minimal random basis of an arbitrary finite matroid whose elements are assigned weights which are independent and uniformly distributed on the interval [0, 1]. This method yields an exact formula in terms of the Tutte polynomial. We give a simple formula to find the minimal random basis of the projective geometry PG (r − 1, q).

We study a polynomial which contains, as special cases, the Tutte polynomials of all members of a given homeomorphism class of graphs. We further show that this polynomial can be directly derived from the chain polynomial introduced in [1]. c © 2002 Elsevier Science B.V. All rights reserved.

In this paper, using a well-known recursion for computing the Tutte polynomial of any graph, we found explicit formulae for the Tutte polynomials of any multi-bridge graph and some 2−tree graphs. Further, several recursive formulae for other graphs such as the fan and the wheel graphs are also discussed.

Let ∆ be a finite sequence of n vectors from a vector space over any field. We consider the subspace of Sym(V) spanned by Q v∈S v, where S is a subsequence of ∆. A result of Orlik and Terao provides a doubly indexed direct sum of this space. The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation T (∆; 1 + x, y). Results of Ardila and Postnikov, Orlik and Terao,...

We prove that any two tilings of a rectangular region by T-tetrominoes are connected by moves involving only 2 and 4 tiles. We also show that the number of such tilings is an evaluation of the Tutte polynomial. The results are extended to a more general class of regions.

The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M , and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte pol...

We study the space X (G) of pictures of a graph G in complex projective d-space. The main result is that the homology groups (with integer coefficients) of X (G) are completely determined by the Tutte polynomial of G. One application is a criterion in terms of the Tutte polynomial for independence in the d-parallel matroids studied in combinatorial rigidity theory. For certain special graphs ca...