Kathryn L. Nyman

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The peak set of a permutation σ is the set {i : σ (i − 1) < σ (i) > σ (i + 1)}. The group algebra of the symmetric group Sn admits a subalgebra in which elements are sums of permutations with a common descent set. In this paper we show the existence of a subalgebra of this descent algebra in which elements are sums of permutations sharing a common peak set.(More)
We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut(More)
In this paper we extend the work of Bogart and Trenk [3] and Fishburn and Trotter [6] in studying different classes of bitolerance orders. We provide a more comprehensive list of classes of bitolerance orders and prove equality between some of these classes in general and other classes in the bipartite domain. We also provide separating examples between(More)
The peak algebra Pn is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of Pn. We discuss two peak analogs of the(More)
The flag Whitney numbers (also referred to as the flag f -numbers) of a geometric lattice count the number of chains of the lattice with elements having specified ranks. We give a collection of inequalities which imply all the linear inequalities satisfied by the flag Whitney numbers of rank 3 geometric lattices. We further describe the smallest closed(More)
We consider an extension of the 2-person Rényi-Ulam liar game in which lies are governed by a channel C, a set of allowable lie strings of maximum length k. Carole selects x ∈ [n], and Paul makes t-ary queries to uniquely determine x. In each of q rounds, Paul weakly partitions [n] = A0∪· · ·∪At−1 and asks for a such that x ∈ Aa. Carole responds with some(More)
We show that Fan’s 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem. Moreover, unlike other Borsuk–Ulam equivalents, we show that this lemma directly implies Sperner’s Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk–Ulam theorem implies the Brouwer(More)
We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f -vector of the(More)