[Paper Review] Generalizations of two-stack-sortable permutations

Dapeng Xu|ArXiv.org|Sep 23, 2002

Advanced Combinatorial Mathematics20 references3 citations

TL;DR

This paper generalizes two-stack-sortable permutations to $r$-permutations, introducing a functional equation for the generating function of two-stack-sortable $k$-tuple $r$-permutations counted by descents using a Zeilberger-style factorization. The solution yields explicit formulas for the number of such permutations, extending classical results on stack-sorting to multiset permutations with applications to pattern-avoidance and symmetric function theory.

ABSTRACT

In this thesis, we apply the stack sorting operator to $r$-permutations and construct the functional equation for the generating function of two-stack-sortable $k$-tuple $r$-permutations counted by descents by using a factorization similar to Zeilberger's. We solve the functional equation and give explicit formulas for the number of two-stack-sortable $r$-permutations.

Motivation & Objective

• To extend the theory of stack-sortable permutations to $r$-permutations, where each element appears $r$ times.
• To enumerate two-stack-sortable $k$-tuple $r$-permutations by descent number using functional equations.
• To generalize West's characterization of 2-stack-sortable permutations to the $r$-permutation setting.
• To explore connections between stack-sorting, pattern avoidance, and symmetric functions in the context of multiset permutations.

Proposed method

• Adapts Zeilberger's functional equation approach to $r$-permutations by introducing a factorization method based on the position of the maximum element.
• Derives a functional equation for the generating function of two-stack-sortable $k$-tuple $r$-permutations using recursive decomposition.
• Applies multivariable Lagrange inversion to extract coefficients from the generating function, yielding explicit formulas.
• Introduces a modification replacing elementary symmetric functions with complete homogeneous symmetric functions to model tree structures with unbounded child types.
• Uses generating functions $G$ and $F$ defined via $G = 1 + zFG$ and $f_k = \sum_{j=0}^\infty g_{k+j} h_j(x)$ to model the combinatorial structure.
• Solves the modified functional equation using substitutions $x_i = \frac{y_i(1 - y_i)}{H^2(y)}$ and derives closed forms for $G$ and $F$.

Experimental results

Research questions

• RQ1 How can the concept of two-stack-sortable permutations be generalized to $r$-permutations where each element appears $r$ times?
• RQ2 What functional equation governs the generating function for two-stack-sortable $k$-tuple $r$-permutations counted by descents?
• RQ3 How does the structure of $r$-permutations affect the pattern-avoidance conditions for two-stack-sortability?
• RQ4 Can the functional equation for $r$-permutations be solved explicitly to yield closed-form expressions for the number of such permutations?
• RQ5 What is the impact of replacing elementary symmetric functions with complete homogeneous symmetric functions on the solution structure of the functional equation?

Key findings

• The number of two-stack-sortable $r$-permutations is given by the formula $\frac{1}{n^2} \prod_{i=0}^{r} \frac{n}{n + k_i} \binom{2n + 2k_i}{k_i}$, where $n = 1 + \sum k_i$.
• An explicit generating function solution is derived using multivariable Lagrange inversion, yielding a closed-form expression for the number of such permutations.
• A modified functional equation using complete homogeneous symmetric functions leads to a new solution form involving $H(y)$, with $G = \frac{c(t)H(y)}{H(y,c(t))}$ and $F = \frac{H(y)}{t}\left(H(y) - \frac{H(y,c(t))}{c(t)}\right)$.
• The solution reveals a duality: $A(-n; k_0, \dots, k_r) = (-1)^{n-1} B(n; k_0, \dots, k_r)$, connecting negative and positive index cases.
• The functional equation framework generalizes to $t$-stack-sortable $\mu$-tuple permutations, though solving it for $t \geq 3$ remains challenging.
• The characterization of two-stack-sortable permutations is extended to $r$-permutations via forbidden patterns: no subsequence of type 2341 or 3241 not part of 35241.