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.
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck's conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher's bijection, we give the combinatorial proofs of these three conjectures directly or by proving more generalized results.
The two partition functions p ω (n) and p ν (n) were introduced by Andrews, Dixit and Yee, which are related to the third order mock theta functions ω(q) and ν(q), respectively. Recently, Andrews and Yee analytically studied two identities that connect the refinements of p ω (n) and p ν (n) with the generalized bivariate mock theta functions ω(z; q) and ν(z; q), respectively. However, they stated these identities cried out for bijective proofs. In this paper, we first define the generalized trivariate mock theta functions ω(y, z; q) and ν(y, z; q). Then by utilizing odd Ferrers graph, we obtain certain identities concerning to ω(y, z; q) and ν(y, z; q), which extend some early results of Andrews that are related to ω(z; q) and ν(z; q). In virtue of the combinatorial interpretations that arise from the identities involving ω(y, z; q) and ν(y, z; q), we finally present bijective proofs for the two identities of Andrews-Yee.
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