Armstrong, Hanusa and Jones conjectured that if s, t are coprime integers, then the average size of an (s, t)-core partition and the average size of a self-conjugate (s, t)core partition are both equal to (s+t+1)(s−1)(t−1) 24. Stanley and Zanello showed that the average size of an (s, s + 1)-core partition equals s+1 3 /2. Based on a bijection of Ford, Mai and Sze between self-conjugate (s, t)-core partitions and lattice paths in ⌊ s 2 ⌋ × ⌊ t 2 ⌋ rectangle, we obtain the average size of a self-conjugate (s, t)-core partition as conjectured by Armstrong, Hanusa and Jones.A partition is of size n if it is a partition of n. Aukerman, Kane and Sze [3] conjectured that the largest size of an (s, t)-core partition for s and t are coprime. Olsson and Stanton [5] proved this conjecture and gave the following stronger theorem.

Given a sequence s = (s 1 , s 2 , . . .) of positive integers, the inversion sequences with respect to s, or s-inversion sequences, were introduced by Savage and Schuster in their study of lecture hall polytopes. A sequence (e 1 , e 2 , . . . , e n ) of nonnegative integers is called an s-inversion sequence of length n if 0 ≤ e i < s i for 1 ≤ i ≤ n. Let I(n) be the set of s-inversion sequences of length n for s = (1, 4, 3, 8, 5, 12, . . .), that is, s 2i = 4i and s 2i−1 = 2i − 1 for i ≥ 1, and let P n be the set of signed permutations on {1 2 , 2 2 , . . . , n 2 }. Savage and Visontai conjectured that when n = 2k, the ascent number over I n is equidistributed with the descent number over P k . For a positive integer n, we use type B P -partitions to give a characterization of signed permutations over which the descent number is equidistributed with the ascent number over I n . When n is even, this confirms the conjecture of Savage and Visontai. Moreover, let I ′ n be the set of s-inversion sequences of length n for s = (2, 2, 6, 4, 10, 6, . . .), that is, s 2i = 2i and s 2i−1 = 4i − 2 for i ≥ 1. We find a set of signed permutations over which the descent number is equidistributed with the ascent number over I ′ n .

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In this paper, we are concerned with counting corners of core partitions. We introduce the concepts of stitches and anti-stitches, which are pairs of cells in a quotient space which we call wrap-up space. We prove that the anti-stitches of a rational Dyck path are in bijection with the segments of structure sets of the corresponding core partition, therefore the corners of a core partition can be counted by the number of stitches or anti-stitches. Based on these results, for coprime positive integers a and b, we give two essentially different formulae for the number of corners in all (a, b)-cores. This leads to an unexpected identity, expressing the rational Catalan numbers as weighed sums of binomial numbers. Moreover, we show that for an (n, n + 1)-core partition λ determined by a certain (n, n + 1)-Dyck path P , the corners of λ correspond to pairs of consecutive right steps in P. As a consequence, we show that the number of (n, n + 1)-cores with k corners is the Narayana number N (n, k + 1). We also extend these results to multi-cores.

Let W be a Coxeter group, and for u, v ∈ W , let R u,v (q) be the Kazhdan-Lusztig R-polynomial indexed by u and v. In this paper, we present a combinatorial proof of the inversion formula on R-polynomials due to Kazhdan and Lusztig. This problem was raised by Brenti. Based on Dyer's combinatorial interpretation of the R-polynomials in terms of increasing Bruhat paths, we reformulate the inversion formula in terms of V -paths. By a V -path from u to v with bottom w we mean a pair (∆ 1 , ∆ 2 ) of Bruhat paths such that ∆ 1 is a decreasing path from u to w and ∆ 2 is an increasing path from w to v. We find a reflection principle on V -paths, which leads to a combinatorial proof of the inversion formula. Moreover, we give two applications of the reflection principle. First, we restrict this involution to V -paths from u to v with maximal length. This provides a direct interpretation for the equi-distribution property that any nontrivial interval [u, v] has as many elements of even length as elements of odd length. This property was obtained by Verma in his derivation of the Möbius function of the Bruhat order. Second, using the reflection principle for the symmetric group, we obtain a refinement of the inversion formula by restricting the summation to permutations ending with a given element.