Anderson established a connection between core partitions and order ideals of certain posets by mapping a partition to its β-set. In this paper, we give a characterization of the poset P (s,s+1,s+2) whose order ideals correspond to (s, s + 1, s + 2)-core partitions. Using this characterization, we obtain the number of (s, s + 1, s + 2)-core partitions, the maximum size and the average size of an (s, s + 1, s + 2)-core partition, confirming three conjectures posed by Amdeberhan.

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.

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In this paper, we present a q-analogue of the polynomial reduction which was originally developed for hypergeometric terms. Using the q-Gosper representation, we describe the structure of rational functions that are summable when multiplied with a given q-hypergeometric term. The structure theorem enables us to generalize the q-polynomail reduction to the rational case, which can be used in the automatic proof and discovery of q-identities. As applications, several q-analogues of series for π are presented.

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