[Paper Review] Partially ordered patterns and compositions

Silvia Heubach, Sergey Kitaev+1 more|ArXiv.org|Oct 1, 2006

Advanced Combinatorial Mathematics16 references8 citations

TL;DR

This paper introduces and analyzes partially ordered patterns (POPs) in integer compositions, extending prior work on pattern avoidance in permutations and words. It derives a q-analogue generating function for the maximum number of non-overlapping occurrences of a segmented POP in compositions, generalizing results from Kitaev (2006) and Kitaev and Mansour (2006) to the composition setting using generating function techniques and multi-pattern avoidance.

ABSTRACT

A partially ordered (generalized) pattern (POP) is a generalized pattern some of whose letters are incomparable, an extension of generalized permutation patterns introduced by Babson and Steingrimsson. POPs were introduced in the symmetric group by Kitaev [Partially ordered generalized patterns, Discrete Math. 298 (2005), 212-229; Introduction to partially ordered patterns, Discrete Appl. Math., to appear], and studied in the set of $k$-ary words by Kitaev and Mansour [Partially ordered generalized patterns and $k$-ary words, Annals of Combinatorics 7 (2003) 191-200]. Moreover, Kitaev et al. [S. Kitaev, T. McAllister and K. Petersen, Enumerating segmented patterns in compositions and encoding with restricted permutations, preprint] introduced segmented POPs in compositions. In this paper, we study avoidance of POPs in compositions and generalize results for avoidance of POPs in permutations and words. Specifically, we obtain results for the generating functions for the number of compositions that avoid shuffle patterns and multi-patterns. In addition, we give the generating function for the distribution of the maximum number of non-overlapping occurrences of a segmented POP $τ$ (that is allowed to have repeated letters) among the compositions of $n$ with $m$ parts in a given set, provided we know the generating function for the number of compositions of $n$ with $m$ parts in the given set that avoid $τ$. This result is a $q$-analogue of the main result in [S. Kitaev, T. Mansour, Partially ordered generalized patterns and $k$-ary words, Annals of Combinatorics 7 (2003) 191-200].

Motivation & Objective

• To extend the theory of pattern avoidance from permutations and words to compositions using partially ordered patterns (POPs).
• To generalize the generating function for the maximum number of non-overlapping occurrences of a pattern in compositions, analogous to results in permutations and words.
• To provide a q-analogue of the main result from Kitaev and Mansour (2006) for words, now applicable to compositions.
• To establish a framework for computing generating functions for compositions avoiding POPs, particularly shuffle patterns and multi-patterns.

Proposed method

• Uses generating functions to model the number of compositions with parts in a given set A that avoid a POP.
• Applies Theorem 3.7 to multi-patterns formed by concatenating a POP τ multiple times, enabling derivation of generating functions for non-overlapping occurrences.
• Introduces a bivariate generating function in variables x (for sum n) and y (for number of parts m), with an additional parameter t tracking the maximum number of non-overlapping occurrences of a pattern.
• Derives a closed-form expression for the generating function of the distribution of non-overlapping POP occurrences via recursive decomposition and bijection arguments.
• Leverages known generating functions for avoidance of a single POP τ to build the full distribution via a rational function expression.
• Applies the method to the specific case of descents (pattern 12), yielding an explicit formula for the maximum number of non-overlapping descents (MND).

Experimental results

Research questions

• RQ1 How can the concept of partially ordered patterns (POPs) be extended to the setting of integer compositions?
• RQ2 What is the generating function for the number of compositions that avoid a given POP, particularly when the POP is a segmented pattern?
• RQ3 Can a q-analogue of the distribution of non-overlapping occurrences of a pattern be derived for compositions, analogous to results in permutations and words?
• RQ4 How does the maximum number of non-overlapping occurrences of a segmented POP τ distribute across compositions with parts in a given set A?
• RQ5 What is the generating function for the maximum number of non-overlapping descents (MND) in compositions with parts in A?

Key findings

• The paper derives a q-analogue generating function for the distribution of the maximum number of non-overlapping occurrences of a segmented POP τ in compositions, generalizing Theorem 5.1 from Kitaev and Mansour (2006) for words.
• For any ordered set A of positive integers and a consecutive pattern τ, the generating function for the distribution of τ-nlap(σ) is given by C_τ^A(x,y) / [1 - t((y∑_{a∈A} x^a - 1)C_τ^A(x,y) + 1)].
• When τ is the descent pattern 12, the generating function for MND is explicitly computed as 1 / [(1−x)(1−x²) − x³t], with a series expansion in t showing the distribution of non-overlapping descents.
• The generating function for compositions avoiding a multi-pattern formed by s copies of τ is expressed recursively using Theorem 3.7, enabling the derivation of the full distribution via generating function manipulation.
• The method successfully generalizes the avoidance framework from permutations and words to compositions, providing a systematic approach to POP avoidance in this new context.
• The result establishes a complete generating function framework for analyzing non-overlapping occurrences of POPs in compositions, with explicit formulas for key cases like descents.