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A signed permutation π = π1π2 . . . πn in the hyperoctahedral group Bn is a word such that each πi ∈ {−n, . . . , −1, 1, . . . , n} and {|π1|, |π2|, . . . , |πn|} = {1, 2, . . . , n}. An index i is a peak of π if πi−1 < πi > πi+1 and PB(π) denotes the set of all peaks of π. Given any set S, we define PB(S, n) to be the set of signed permutations π ∈ Bn with PB(π) = S. In this paper we are interested in the cardinality of the set PB(S, n). In [4], Billey, Burdzy and Sagan investigated the analogous problem for permutations in the symmetric group, Sn. In this paper we extend their results to the hyperoctahedral group; in particular we show that #PB(S, n) = p(n)2 2n−|S|−1 where p(n) is the same polynomial found in [4] which leads to the explicit computation of interesting special cases of the polynomial p(n). In addition we have extended these results to the case where we add π0 = 0 at the beginning of the permutations, which gives rise to the possibility of a peak at position 1, for both the symmetric and the hyperoctahedral groups.

An arithmetical structure on a finite, connected graph G is a pair of vectors (d, r) with positive integer entries for which (diag(d) − A)r = 0, where A is the adjacency matrix of G and where the entries of r have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of (diag(d) − A). In this paper, we study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two "prongs" at one end. We give a process for determining the number of arithmetical structures on the bident with n vertices and show that this number grows at the same rate as the Catalan numbers as n increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.Let G be a finite, connected graph with n vertices, and let A be the adjacency matrix of G. An arithmetical structure on G is given by a pair of vectors (d, r) ∈ (Z >0 ) n × (Z >0 ) n for which the

We say that a permutation π = π 1 π 2 · · · π n ∈ S n has a peak at index i if π i−1 < π i > π i+1 . Let P(π) denote the set of indices where π has a peak. Given a set S of positive integers, we define P S (n) = {π ∈ S n : P(π) = S}. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large n, |P S (n)| = p S (n)2 n−|S|−1 where p S (x) is a polynomial depending on S. They gave a recursive formula for p S (x) involving an alternating sum, and they conjectured that the coefficients of p S (x) expanded in a binomial coefficient basis centered at max(S) are all nonnegative. In this paper we introduce a new recursive formula for |P S (n)| without alternating sums, and we use this recursion to prove that their conjecture is true.

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