Three families of pattern-avoiding ascent sequences and generating trees*

Ronit Mansour; Armend Sh Shabani

doi:10.26493/2590-9770.1952.f7b

Three families of pattern-avoiding ascent sequences and generating trees^*

Ronit Mansour
, Armend Sh Shabani

Research output: Contribution to journal › Article › peer-review

Abstract

An ascent sequence a1a2 · · · an is one consisting of non-negative integers satisfying ai =0 and ai < asc(a1a2 …ai-1) + 1 for 1 < i < n, where asc(a1a2 …ak)denotes the number of ascents in a1a2 · · · ak. In this work, we provide alternative proofs and extensions of two known enumeration results: namely, the number of ascent sequences of length n that avoid the pattern 0012 is given by the nth Catalan number, whereas those avoiding 0112 are counted by (3 ^n-1 + 1)/2. Furthermore, we investigate the generating function associated with ascent sequences avoiding the pattern 0122.

Original language	English
Journal	Art of Discrete and Applied Mathematics
DOIs	https://doi.org/10.26493/2590-9770.1952.f7b
State	Published - Jan 2026
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2026 This work is licensed under https://creativecommons.org/licenses/by/4.0/

Keywords

Ascent sequences
Catalan number
generating trees
Kernel method

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

Access to Document

10.26493/2590-9770.1952.f7b

Cite this

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title = "Three families of pattern-avoiding ascent sequences and generating trees*",

abstract = "An ascent sequence a1a2 · · · an is one consisting of non-negative integers satisfying ai =0 and ai < asc(a1a2 …ai-1) + 1 for 1 < i < n, where asc(a1a2 …ak)denotes the number of ascents in a1a2 · · · ak. In this work, we provide alternative proofs and extensions of two known enumeration results: namely, the number of ascent sequences of length n that avoid the pattern 0012 is given by the nth Catalan number, whereas those avoiding 0112 are counted by (3 n-1 + 1)/2. Furthermore, we investigate the generating function associated with ascent sequences avoiding the pattern 0122.",

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doi = "10.26493/2590-9770.1952.f7b",

language = "English",

journal = "Art of Discrete and Applied Mathematics",

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AU - Shabani, Armend Sh

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N2 - An ascent sequence a1a2 · · · an is one consisting of non-negative integers satisfying ai =0 and ai < asc(a1a2 …ai-1) + 1 for 1 < i < n, where asc(a1a2 …ak)denotes the number of ascents in a1a2 · · · ak. In this work, we provide alternative proofs and extensions of two known enumeration results: namely, the number of ascent sequences of length n that avoid the pattern 0012 is given by the nth Catalan number, whereas those avoiding 0112 are counted by (3 n-1 + 1)/2. Furthermore, we investigate the generating function associated with ascent sequences avoiding the pattern 0122.

AB - An ascent sequence a1a2 · · · an is one consisting of non-negative integers satisfying ai =0 and ai < asc(a1a2 …ai-1) + 1 for 1 < i < n, where asc(a1a2 …ak)denotes the number of ascents in a1a2 · · · ak. In this work, we provide alternative proofs and extensions of two known enumeration results: namely, the number of ascent sequences of length n that avoid the pattern 0012 is given by the nth Catalan number, whereas those avoiding 0112 are counted by (3 n-1 + 1)/2. Furthermore, we investigate the generating function associated with ascent sequences avoiding the pattern 0122.

KW - Ascent sequences

KW - Catalan number

KW - generating trees

KW - Kernel method

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