[Paper Review] A q-enumeration of alternating permutations

Matthieu Josuat-Vergès|arXiv (Cornell University)|Jul 6, 2009

Advanced Combinatorial Mathematics6 references4 citations

TL;DR

This paper presents a q-enumeration of alternating permutations using the number of crossings in permutations and derangements, refining classical Euler and Roselle identities. It derives closed-form formulas for q-tangent and q-secant numbers via two methods: permutation tableaux and weighted Motzkin paths (Laguerre histories), establishing a connection between q-Eulerian statistics and continued fraction expansions of q-tangent and q-secant generating functions.

ABSTRACT

A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both identities can be refined with the following statistics: the number of crossings in permutations and derangements, and the number of patterns 31-2 in alternating permutations. Using previous results of Corteel, Rubey, Prellberg, and the author, we derive closed formulas for both q-tangent and q-secant numbers. There are two different methods to obtain these formulas: one with permutation tableaux and one with weighted Motzkin paths (Laguerre histories).

Motivation & Objective

• To refine classical Euler and Roselle identities on tangent and secant numbers using q-analogs.
• To establish a connection between q-tangent and q-secant numbers and the statistics of crossings and weak exceedances in permutations and derangements.
• To derive closed-form expressions for q-tangent and q-secant numbers using combinatorial methods.
• To unify two distinct combinatorial approaches—permutation tableaux and weighted Motzkin paths—for proving these identities.

Proposed method

• The paper uses the statistic of crossings in permutations, denoted cr(σ), and weak exceedances, wex(σ), to refine signed sums over symmetric and derangement groups.
• It applies the theory of continued fractions to express q-tangent and q-secant generating functions as T-fractions with q-integers [k]_q.
• The method involves constructing weighted Dyck and Schröder paths where step weights depend on height and are assigned powers of q.
• It employs generating functions involving basic hypergeometric series 2φ1 and Heine’s transformation to evaluate path sums.
• Two independent proofs are provided: one using permutation tableaux and the other using Laguerre histories (weighted Motzkin paths).
• The closed formulas are derived by decomposing path weights and applying sign-reversing involutions or coefficient extraction from hypergeometric series.

Experimental results

Research questions

• RQ1 Can the classical Euler and Roselle identities for tangent and secant numbers be refined using q-analogs of permutation statistics like crossings and weak exceedances?
• RQ2 What is the combinatorial interpretation of the q-tangent and q-secant numbers in terms of permutation tableaux and weighted lattice paths?
• RQ3 How can the signed sums over permutations and derangements involving wex(σ) and cr(σ) be related to the continued fraction expansions of q-tangent and q-secant generating functions?
• RQ4 Is there a unified parity-independent formula for E_n(q) that combines both q-tangent and q-secant cases?
• RQ5 Can the closed-form expressions for E_n(q) be derived via sign-reversing involutions on weighted paths?

Key findings

• The paper establishes that ∑σ∈𝔖n (−1)^wex(σ) q^cr(σ) equals (−1)^((n−1)/2) E_n(q) when n is odd and 0 when n is even.
• For derangements, ∑σ∈𝒟_n (−1/q)^wex(σ) q^cr(σ) equals (−1/q)^(n/2) E_n(q) when n is even and 0 when n is odd.
• Closed-form formulas for E_{2n+1}(q) and E_{2n}(q) are derived using binomial coefficients and alternating sums over q-exponents.
• The formula for E_{2n+1}(q) is given by (1−q)^{−(2n+1)} ∑_{k=0}^n (binom{2n+1}{n−k} − binom{2n+1}{n−k−1}) ∑_{i=0}^{2k+1} (−1)^{i+k} q^{i(2k+2−i)}
• The formula for E_{2n}(q) is given by (1−q)^{−2n} ∑_{k=0}^n (binom{2n}{n−k} − binom{2n}{n−k−1}) ∑_{i=0}^{2k} (−1)^{i+k} q^{i(2k−i)+k}
• A unified expression for E_n(q) is derived as (−1)^{⌊n/2⌋}/(1−q)^n times a sum over k and i involving binomial differences and q-exponents, valid for both even and odd n.