Bijections between t-core partitions and t-tuples

T-partitions and S-complete T-partitions of a Graph

Bijections between noncrossing and nonnesting partitions for classical reflection groups

We present type preserving bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types B, C and D are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspon...

Three Bijections on Set Partitions

We study three similar bijections on set partitions. The first gives a bijective proof of the equivalence of two statistics with a q-Stirling distribution, Milne’s statistic and the intertwining number. The second proves the equivalence of a multivariate block size distribution to a covering statistic. The third demonstrates equivalence of the number of all set partitions up to a given size to ...

Some Bijections on Set Partitions

We study three similar bijections on set partitions. The first is an involution defined by Kasraoui and Zeng which proves the symmetry of the distribution of crossings and nestings. We show that a stronger result can be deduced. The second gives a bijective proof of the equivalence of two statistics with a q-Stirling distribution. The third proves the equivalence of a multivariate block size di...

Cayley Compositions, Partitions, Polytopes, and Geometric Bijections

In 1857, Cayley showed that certain sequences, now called Cayley compositions, are equinumerous with certain partitions into powers of 2. In this paper we give a simple bijective proof of this result and a geometric generalization to equality of Ehrhart polynomials between two convex polytopes. We then apply our results to give a new proof of Braun’s conjecture proved recently by the authors [K...