A skeleton model to enumerate standard puzzle sequences

<abstract><p>Guo-Niu Han [Sémin. Lothar. Comb. 85 (2021) B85c (electronic)] has introduced a new combinatorial object named standard puzzle. We use digraphs to show the relations between numbers in standard puzzles and propose a skeleton model. By this model, we solve the enumeration problem of over fifty thousand standard puzzle sequences. Most of them can be represented by classical numbers, such as Catalan numbers, double factorials, secant numbers and so on. Also, we prove several identities for standard puzzle sequences.</p></abstract>

Combinatorics of geometrically distributed random variables: newq-tangent andq-secant numbers

Up-down permutations are counted by tangent (respectively, secant) numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all coincide with the classical version. In this way, we get some newq-tangent andq-secant functions. Some of them also have nice continued fraction expansions; in one particular case, we could not find a proof for it. Divisibility results à la Andrews, Foata, Gessel are also discussed.

Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences

Let $P_n$ and $Q_n$ be the polynomials obtained by repeated differentiation of the tangent and secant functions respectively. From the exponential generating functions of these polynomials we develop relations among their values, which are then applied to various numerical sequences which occur as values of the $P_n$ and $Q_n$. For example, $P_n(0)$ and $Q_n(0)$ are respectively the $n$th tangent and secant numbers, while $P_n(0)+Q_n(0)$ is the $n$th André number. The André numbers, along with the numbers $Q_n(1)$ and $P_n(1)-Q_n(1)$, are the Springer numbers of root systems of types $A_n$, $B_n$, and $D_n$ respectively, or alternatively (following V. I. Arnol'd) count the number of "snakes" of these types. We prove this for the latter two cases using combinatorial arguments. We relate the values of $P_n$ and $Q_n$ at $\sqrt3$ to certain "generalized Euler and class numbers" of D. Shanks, which have a combinatorial interpretation in terms of 3-signed permutations as defined by R. Ehrenborg and M. A. Readdy. Finally, we express the values of Euler polynomials at any rational argument in terms of $P_n$ and $Q_n$, and from this deduce formulas for Springer and Shanks numbers in terms of Euler polynomials.