Let Q n denote the n-dimensional hypercube with the vertex set V n = {0, 1} n . A 0/1-polytope of Q n is a convex hull of a subset of V n . This paper is concerned with the enumeration of equivalence classes of full-dimensional 0/1-polytopes under the symmetries of the hypercube. With the aid of a computer program, Aichholzer completed the enumeration of equivalence classes of full-dimensional 0/1-polytopes for Q 4 , Q 5 , and those of Q 6 up to 12 vertices. In this paper, we present a method to compute the number of equivalence classes of full-dimensional 0/1-polytopes of Q n with more than 2 n−3 vertices. As an application, we finish the counting of equivalence classes of full-dimensional 0/1-polytopes of Q 6 with more than 12 vertices.

Motivated by the question of finding a type B analogue of the bijection between oscillating tableaux and matchings, we find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m-rim hook tableau is defined as a sequence (λ 0 , λ 1 , . . . , λ 2n ) of Young diagrams starting with the empty shape and ending with the empty shape such that λ i is obtained from λ i−1 by adding an mrim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook tableau is also called an oscillating domino tableau. When we restrict our attention to two column oscillating domino tableaux of length 2n, we are led to a bijection between such tableaux and noncrossing 2-colored matchings on {1, 2, . . . , 2n}, which are counted by the product C n C n+1 of two consecutive Catalan numbers. A 2-colored matching is noncrossing if there are no two arcs of the same color that are intersecting. We show that oscillating domino tableaux with at most two columns are in one-to-one correspondence with Dyck path packings. A Dyck path packing of length 2n is a pair (D, E), where D is a Dyck path of length 2n, and E is a dispersed Dyck path of length 2n that is weakly covered by D. So we deduce that Dyck path packings of length 2n are counted by C n C n+1 .

A signed labeled forest is defined as a (plane) forest labeled by {1, 2, . . . , n} along with minus signs associated to some vertices. Signed labeled forests can be viewed as an extension of signed permutations. We define the inversion number, the flag major index and the R-major index on signed labeled forests. They can be considered as type B analogues of the indices for labeled forests introduced by Björner and Wachs. The flag major index for signed labeled forests is based on the flag major index on signed permutations introduced by Adin and Roichman, whereas the R-major index for signed labeled forests is based on the R-major index that we introduce for signed permutations, which is closely related to the major defined by Reiner. We obtain q-hook length formulas by q-counting signed labelings of a given forest with respect to the above indices, from which we see that these three indices are equidistributed for signed labeled forests. Our formulas for the major indices and the inversion number are type B analogues of the formula due to Björner and Wachs. We also give a type D analogue with respect to the inversion number of even-signed labeled forests.

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 .

Using equivariant cohomology theory, Naruse obtained a hook length formula for the number of standard Young tableaux of skew shape λ/µ. Morales, Pak and Panova found two q-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape λ/µ and reverse plane partitions of shape λ/µ. When λ and µ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape λ/µ can be expressed as a determinant whose entries are related to q-analogues of the Euler numbers. The objective of this paper is to prove this conjecture.

Recently Han obtained a general formula for the weight function corresponding to the expansion of a generating function in terms of hook lengths of binary trees. In this paper, we present formulas for k-ary trees, plane trees, plane forests, labeled trees and forests. We also find appropriate generating functions which lead to unifications of the hook length formulas due to Du and Liu, Han, Gessel and Seo, and Postnikov.

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Gustafson and Milne proved an identity on the Schur function indexed by a partition of the form (λ 1 − n + k, λ 2 − n + k, . . . , λ k − n + k). On the other hand, Fehér, Némethi and Rimányi found an identity on the Schur function indexed by a partition of the form (m − k, . . . , m − k, λ 1 , . . . , λ k ). Fehér, Némethi and Rimányi gave a geometric explanation of their identity, and they raised the question of finding a combinatorial proof. In this paper, we establish a Gustafson-Milne type identity as well as a Fehér-Némethi-Rimányi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Fehér-Némethi-Rimányi identity.