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.

As shown by Bousquet-Mélou-Claesson- Dukes-Kitaev (2010), ascent sequences can be used to encode (2 + 2)-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power seriesIn this paper, we present a novel way to recursively decompose ascent sequences, which leads to:• a calculation of the Euler-Stirling distribution on ascent sequences, including the numbers of ascents (asc), repeated entries (rep), zeros (zero) and maximal entries (max).In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of zero and max. • a far-reaching generalization of the generating function formula for (asc, zero) due toJelínek. This is accomplished via a bijective proof of the quadruple equidistribution of (asc, rep, zero, max) and (rep, asc, rmin, zero), where rmin denotes the right-to-left minima statistic of ascent sequences. • an extension of a conjecture posed by Levande, which asserts that the pair (asc, zero) on ascent sequences has the same distribution as the pair (rep, max) on (2 − 1)-avoiding inversion sequences. This is achieved via a decomposition of (2 − 1)-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples (asc, rep, zero, max) and (rep, asc, max, zero) are equidistributed on ascent sequences.

Abstract. In this paper, we are mainly concerned with the enumeration of (2k + 1, 2k + 3)-core partitions with distinct parts. We derive the number and the largest size of such partitions, confirming two conjectures posed by Straub.

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An ordered partition of [n] = {1, 2, . . . , n} is a partition whose blocks are endowed with a linear order. Let OP n,k be set of ordered partitions of [n] with k blocks and OP n,k (σ) be set of ordered partitions in OP n,k that avoid a pattern σ. Recently, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of ordered partitions of [n] with 3 blocks and the number of ordered partitions of [n] with n − 1 blocks avoiding a permutation pattern of length 3. They showed that |OP n,k (σ)| = |OP n,k (123)| for any permutation σ of length 3, and raised the question concerning the enumeration of OP n,k (123). They also conjectured that the number of ordered partitions of [2n] with blocks of size 2 avoiding a permutation pattern of length 3 satisfied a second order linear recurrence relation.In answer to the question of Godbole, et al., we obtain the generating function for |OP n,k (123)| and we prove the conjecture on the recurrence relation.