Cyclotomic factors of the descent set polynomial

Abstract

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up notatio

On the powers of the descent set statistic

“…Some of the results in this paper are reminiscent of results in the papers [1,2,3], where there are results which depend on the binary expansion of the parameters. However, as the reader can see from Tables 1 and 2, where we present computational results for the numbers of factors of the primes 2 and 3 in A r n , a lot of work remains in order to understand these numbers.…”

“…. , c k ), where c i = s i − s i−1 with s 0 = 0 and s k = n. See, for instance, [1] or [7,Section 7.19]. It is now straightfoward to observe that α n (S) is given by the multinomial coefficient n co(S) .…”

The descent set polynomial revisited

“…Chebikin, Ehrenborg, Pylyavskyy and Readdy [3] defined the nth descent set polynomial to be Q n (t) = S⊆[n −1] t βn(S) .…”

“…The purpose of quasi-symmetric functions is that they allow efficient computations of the flag f -vector modulo a prime p, using the classical relation (x + y) p ≡ x p + y p mod p. Finally, using the inclusionexclusion equation (2.1), we obtain information about the descent set statistics. Below is a lemma, adapted from Lemma 3.2 in [3], to compute the quasi-symmetric function of the Boolean algebra F (B n ) = M n (1) modulo a prime.…”

“…The following are consequences of Theorem 2.1 in [3], which gives information about the proportion of even or odd descent statistics β n (S) depending on the number of 1's in the binary expansion of n. We apply their result to achieve equalities involving a i,j .…”