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In his recent work Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions. In this note we provide bijective proofs as well as refinements of those three theorems of Andrews. Our refinements take into account the numbers of parts in each of the two colors. [ABSTRACT FROM AUTHOR]

We derive some new signed Mahonian polynomials over the complex reflection group G (r , 1 , n) = C r ≀ S n , where the "sign" is taken to be any of the 2r 1-dim characters and the "Mahonian" statistics are the lmaj defined by Bagno and the sor defined by Eu et al. Various new signed Mahonian polynomials over Coxeter groups of types B n and D n are obtained as well. We also investigate the signed counting polynomials on G(r, 1, n) for those statistics with the distribution [ r ] q [ 2 r ] q ⋯ [ n r ] q . [ABSTRACT FROM AUTHOR]

Motivated by Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchy's identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan. [ABSTRACT FROM AUTHOR]

For an integer m ≥ 2 , a partition λ = (λ 1 , λ 2 , ...) is called m-falling, a notion introduced by Keith, if the least non-negative residues mod m of λ i 's form a nonincreasing sequence. We extend a bijection originally due to the third author to deduce a lecture hall theorem for such m-falling partitions. A special case of this result gives rise to a finite version of Pak–Postnikov's (m, c)-generalization of Euler's theorem. Our work is partially motivated by a recent extension of Euler's theorem for all moduli, due to Xiong and Keith. We note that their result actually can be refined with one more parameter. [ABSTRACT FROM AUTHOR]

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for k-colored partition functions p-k(n) for all k≥2. This enables us to extend the k-colored partition function multiplicatively to a function on k-colored partitions and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results. [ABSTRACT FROM AUTHOR]

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's Odd-Distinct partition theorem, derive a generalization in the spirit of Alder's conjecture, as well as a curious analogue of the first Rogers-Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler's pentagonal number theorem in this setting, and connect it with the Rogers-Fine identity. We conclude with some congruence properties. [ABSTRACT FROM AUTHOR]