A Tutte decomposition for matrices and bimatroids

AbstractWe develop a Tutte decomposition theory for matrices and their combinatorial abstractions, bimatroids. As in the graph or matroid case, this theory is based on a deletion–contraction decomposition. The contribution from the deletion, derived by an inclusion–exclusion argument, consists of three terms. With one more term contributed from the contraction, the decomposition has four terms in general. There are universal decomposition invariants, one of them being a corank–nullity polynomial. Under a simple change of variables, the corank–nullity polynomial equals a weighted characteristic polynomial. This gives an analog of an identity of Tutte. Applications to counting and critical problems on matrices and graphs are given

Minimal blocks of binary even-weight vectors

AbstractOdd circuits are minimal 1-blocks over GF(2) and the odd circuit of size 2t+1 can be represented by the vectors of Hamming weight 2t in a (2t+1)-dimensional vector space over GF(2). This is the tip of an iceberg. Let f(2t,k,2) be the maximum number of binary k-dimensional column vectors such that for all s, 1⩽s⩽t, no 2s columns sum to the zero vector. If k=2, k=3, k=4, or k⩾5 and 2t is sufficiently large (for example, 2t⩾2k−k+1 suffices), then the set of vectors of weight 2t in a (f(2t,k,2)+2t−1)-dimensional vector space over GF(2) is a minimal k-block over GF(2)

Minimal blocks of points with weight divisible by p over GF(p)

AbstractWe construct three families of minimal blocks over GF(p) where p is an odd prime. For example, we show that the points in rank-(2p−1) projective space PG(2p−2,p) with p coordinates equal to 1 and p−1 coordinates equal to 0 form a minimal 1-block over GF(p). The proofs use the Chevalley–Warning theorem about the number of zeros of polynomials over finite fields

Gonc̆arov polynomials and parking functions

AbstractLet u be a sequence of non-decreasing positive integers. A u-parking function of length n is a sequence (x1,x2,…,xn) whose order statistics (the sequence (x(1),x(2),…,x(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x(i)⩽ui. The Gonc̆arov polynomials gn(x;a0,a1,…,an−1) are polynomials defined by the biorthogonality relation:ε(ai)Dign(x;a0,a1,…,an−1)=n!δin,where ε(a) is evaluation at a and D is the differentiation operator. In this paper we show that Gonc̆arov polynomials form a natural basis of polynomials for working with u-parking functions. For example, the number of u-parking functions of length n is (−1)ngn(0;u1,u2,…,un). Various properties of Gonc̆arov polynomials are discussed. In particular, Gonc̆arov polynomials satisfy a linear recursion obtained by expanding xn as a linear combination of Gonc̆arov polynomials, which leads to a decomposition of an arbitrary sequence of positive integers into two subsequences: a “maximum” u-parking function and a subsequence consisting of terms of higher values. Many counting results for parking functions can be derived from this decomposition. We give, as examples, formulas for sum enumerators, and a linear recursion and Appell relation for factorial moments of sums of u-parking functions

The long-line graph of a combinatorial geometry. II. Geometries representable over two fields of different characteristics

AbstractLet q be a power of a prime and let s be zero or a prime not dividing q. Then the number of points in a combinatorial geometry (or simple matroid) of rank n which is representable over GF(q) and a field of characteristic s is at most (qν − qν−1)(2n+1)−n, where ν = 2q−1 − 1

Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries

Let q be an odd prime power not divisible by 3. In Part I of this series, it was shown that the number of points in a rank-n combinatorial geometry (or simple matroid) representable over GF(3) and GF(q) is at most n2. In this paper, we show that, with the exception of n = 3, a rank-n geometry that is representable over GF(3) and GF(q) and contains exactly n2 points is isomorphic to the rank-n Dowling geometry based on the multiplicative group of GF(3). © 1988 Springer-Verlag