scholarly journals Two Other Interpretations of the Entringer Numbers

1997 ◽  
Vol 18 (8) ◽  
pp. 939-943 ◽  
Author(s):  
Christiane Poupard
Keyword(s):  
Entringer Numbers

scholarly journals Extensions of Partial Cyclic Orders, Euler Numbers and Multidimensional Boustrophedons

10.37236/7145 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sanjay Ramassamy
Keyword(s):  
Cyclic Order ◽  
Euler Numbers ◽  
Entringer Numbers

We enumerate total cyclic orders on $\left\{x_1,\ldots,x_n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(x_i,x_{i+1},x_{i+2})$, with indices taken modulo $n$. In some cases, the problem reduces to the enumeration of descent classes of permutations, which is done via the boustrophedon construction. In other cases, we solve the question by introducing multidimensional versions of the boustrophedon. In particular we find new interpretations for the Euler up/down numbers and the Entringer numbers.

scholarly journals Seidel triangle sequences and Bi-Entringer numbers

2014 ◽  
Vol 42 ◽  
pp. 243-260 ◽  
Author(s):  
Dominique Foata ◽  
Guo-Niu Han
Keyword(s):  
Entringer Numbers

scholarly journals The Complete cd-Index of Boolean Lattices

10.37236/5100 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Neil J.Y. Fan ◽  
Liao He
Keyword(s):  
Coxeter Group ◽  
Nonnegative Integer ◽  
Boolean Lattice ◽  
Combinatorial Interpretation ◽  
Euler Numbers ◽  
Combinatorial Interpretations ◽  
Bruhat Graph ◽  
Boolean Lattices ◽  
One To One ◽  
Entringer Numbers

Let $[u,v]$ be a Bruhat interval of a Coxeter group such that the Bruhat graph $BG(u,v)$ of $[u,v]$ is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of $[u,v]$. Since in this case the complete cd-index and the cd-index of $[u,v]$ coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in $BG(u,v)$ by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in $BG(u,v)$. Based on this labeling, we construct a flip $\mathcal{F}$ on the set of Bruhat paths in $BG(u,v)$, which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip $\mathcal{F}$ is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial $M$. Thus by results of Karu, the coefficient of $M$ enumerates certain Bruhat paths in $BG(u,v)$, and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip $\mathcal{F}$. We enumerate the number of cd-monomials in the complete cd-index of $[u,v]$ in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d${}^n$ in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.

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