On the Nonnegativity of the Coefficients of Some Polynomials Occurring in the Theory of Cubical Spheres

As part of his recent research into Cubical Spheres, G. Hetyei gave an inductive proof that a certain sequence of polynomials have nonnegative coe cients. In a poster session at the FPSAC'96 conference he asked for a combinatorial interpretation of these coe cients. In this short note we show that known results in rook theory imply that the coe cients count permutations subject to certain constraints. R esum e. Dans un de ces derniers articles portant sur les sph eres cubiques, G. Hetyei a donn e une preuve par r ecurrence du fait que les coe cients d'une certaine suite de polynômes sont non-n egatifs. A l'occasion du colloque SFCA'96 G. Hetyei a propos e de trouver une interpr etation combinatoire de ces coe cients. Dans ce r esum e on d emontre que certains r esultats connus sur la th eorie des tours impliquent que les dits coe cients comptent bien des permutations sujettes a certaines contraintes. At the recent FPSAC '96 conference held at the University of Minnesota, G. Hetyei presented a poster session entitled \Invariants of Cubical Spheres" [Het1], a full length version of which appeared in this journal [Het2]. The following lemma was part of his session. Lemma Let a;b;c(x) denote the formal power series P k 0 k (k + 1)(k + 2)x, where a; b and c are natural numbers. Then the following hold. (i) We have a;b;c(x) = Aa;b;c(x) (1 x)a+b+c+1 ; (1) where Aa;b;c(x) is a polynomial with integer coe cients, of degree at most a+ b+ c. (ii) If b > 0 then the degree of Aa;b;c(x) is at most a+ b+ c 1. (iii) If b > 0 or c = 0 then Aa;b;c(x) has only nonnegative coe cients. Hetyei's proof used induction and di erential equations satis ed by a;b;c(x). As a challenge he asked for a combinatorial interpretation of part (iii) of his lemma. In this note we use rook theory to show that if b > 0 or c = 0, the coe cients of Aa;b;c(t) count permutations subject to certain constraints. Typeset by AMS-TEX