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.
We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.
This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an algorithm to find Ramanujan-type identities for a(mn + t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n + 6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions p(5n + 2) and p(5n + 3) and Andrews-Paule's broken 2-diamond partition functions △ 2 (25n + 14) and △ 2 (25n + 24). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews' singular overpartition functions Q 3,1 (9n + 3) and Q 3,1 (9n + 6) due to Shen, the 2-dissection formulas of Ramanujan and the 8-dissection formulas due to Hirschhorn.Atkin and Swinnerton-Dyer [5] have shown that g t can always be expressed by certain infinite products for m > 3. Then the left hand side of (1.3) can be expressed in terms of certain infinite products. Kolberg pointed out that when m > 5, this becomes much more complicated. For m = 11, 13, Bilgici and Ekin [7,8] used the method of Kolberg to compute the generating function ∞ n=0 p(mn + t)q mn+t 1. M |N .
Let p k (n) be given by the k-th power of the Euler ProductBy investigating the properties of the modular equations of the second and the third order under the Atkin U -operator, we determine the generating functions of p 8k (2) in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo m, we obtain infinite families of congruences for p k (n) modulo any m ≥ 2, where 1 ≤ k ≤ 24 and 3|k or 8|k. Based on these congruences for p k (n), infinite families of congruences for many partition functions such as the overpartition function, t-core partition functions and ℓ-regular partition functions are easily obtained.
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