By M. Deza and I.G. Rosenberg (Eds.)
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Extra resources for Combinatorics 79 Part I
We interpret the coefficients in the homogeneous polynomials f: and f t - l f i + l . Each term of either is an integer multiple of a monomial of the form g= n n Xi’ X , E Y x;, X,EZ where Y and Z are disjoint subsets of X satisfying lY(+ 2 ( Z (= 21. Let (YI = 2k, so The independent set numbers of a finite matroid 23 121= 1 - k. Then the coefficient of g in f: is the number of ordered pairs (A U Z, €3 U 2 )of independent sets in the restriction M ( Y U Z ) such that (A, €3) is an ordered partition of Y and \ A \= IBI = k.
5 there is a characterization by excluded minors of a-minimal matroid perspectives. The list of excluded minors is infinite: for all n 3 3 (ff :-I, ‘5L) is not a-minimal but every proper minor is. e. has a binary major, if and only if M and M’ are binary and ( M , M ’ ) has no minor isomorphic to one of (‘5::+1, ‘ 5 i k + l ) k 3 1 (see ). It follows that an a-minimal matroid perspective is binary. Furthermore, using Brylawski’s characterizations of series-parallel matroids [ 3 ] , it can be shown that if ( M , M ‘ ) is a-minimal, then M and M’ are series-parallel matroids.
There are a number of interesting conjectures about these sequences. Rota conjectured that the sequence (W,) of Whitney numbers is unimodal, and there is evidence to support the stronger conjecture that it is logarithmically concave. Another conjecture is that wk G wr-kwhen k G r - k. This is true for k = 1 (see [l,4,6]), but in general we have the weaker inequality [ 3 ] W , + W 2 + . . +Wr-k. r Another inequality concerning the (W,) sequence appears in . Analogous conjectures have been made for the sequence (1,) of independent set numbers.