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In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each class and is given by the entries of a new Schroder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim. We employ a variety of techniques to prove our results, including generating trees, direct bijections and the kernel method. For the latter, we make use of in a creative way what we are trying to show in three cases to aid in solving a system of functional equations satisfied by the associated generating functions.
Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schroder paths (resp. little Schroder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: begin{align*} r(n,0)&=sumlimits_{j=0}^{
Let $G=(V(G), E(G))$ be a multigraph with maximum degree $Delta(G)$, chromatic index $chi(G)$ and total chromatic number $chi(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $chi(G)leq Delta(G)+mu(G) +1$ f
We study the generating function of descent numbers for the permutations with descent pairs of prescribed parities, the distribution of which turns out to be a refinement of median Genocchi numbers. We prove the $gamma$-positivity for the polynomial
There is a bijection from Schroder paths to {4132, 4231}-avoiding permutations due to Bandlow, Egge, and Killpatrick that sends area to inversion number. Here we give a concise description of this bijection.
Karasev conjectured that for any set of $3k$ lines in general position in the plane, which is partitioned into $3$ color classes of equal size $k$, the set can be partitioned into $k$ colorful 3-subsets such that all the triangles formed by the subse