Marie-Pierre Aubry, Kenneth G. Miller, Elena Turco, José Abel Flores, Andrey Gladenkov, Patrick Grunert, Frits Hilgen, Hiroshi Nishi, Ann Holbourn, Wout Krijgsman, Fabrizio Lirer, Werner E. Piller, Frédéric Quillévéré, Isabella Raffi, Marci Robinson, Lorenzo Rook, Jun Tian, Maria Triantaphyllou, Felipe Vallejo. Episodes 2022;45:445-53. https://doi.org/10.18814/epiiugs/2022/022008

Since the 1980s, a significant decline in number of breeding populations Rook, Corvus frugilegus Linnaeus, 1758, has been observed many European countries. In Ukraine, this trend since late 1990s. 2021, conservation status Rook Europe was significantly upgraded — from “LC” (Least Concern) to “VU” (Vulnerable). The analysis material surveys 1983–1985 and 2021 on territory 1850 km² Kyiv Region sh...

Let an count the number of 2-dimensional rook paths Rn,n from (0, 0) to (2n, 0). Rook pathsRm,n are the lattice paths from (0, 0) to (m+n,m−n) with allowed steps (x, x) and (y,−y) where x, y ∈ N+. In answer to the open question proposed by M. Erickson et al. (2010), we shall present a combinatorial proof for the recurrence of an, i.e., (n + 1)an+1 + 9(n − 1)an−1 = 2(5n + 2)an with initial condi...

Abstract. Given two operators M and N subject to the relation MN −qNM = p, and a word w in M and N , the rewriting of w in normal form is combinatorially described by rook placements in a Young diagram. We give enumerative results about these rook placements, particularly in the case where p = (1−q)/q. This case naturally arises in the context of the PASEP, a random process whose partition func...

The simplicial rook graph SR(m,n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that SR(m,n) has integral eigenvalues, and smallest eigenvalue s = max(−n,− ( m 2 ) ), and that this graph has a large part of its spectrum in common with the Johnson graph...

We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found these bases, including Eulerian numbers, Stirling quasi-Yamanouchi tableaux, and rook boards. These also lead further about fundamental quasisymmetric expansions which we prove special cases.

In this paper, we shall construct a bijection between rook placements on double staircases (introduced by Josuat-Vergès in 2017) and increasing binary trees. We introduce two subclasses of staircases, which call left right-aligned placements. show that their enumeration, while keeping track certain statistic, gives the $\gamma$-vectors Eulerian polynomials. conclude with discussion different fi...