Counting staircases in integer compositions

Blecher Aubrey; Mansour Toufik

Counting staircases in integer compositions

Department of Mathematics

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

Abstract

The main theorem establishes the generating function F which counts the number of times the staircase 1⁺2⁺3⁺⋯m⁺ fits inside an integer composition of n. F = k_m - qx^my/1-x k_m-1/(1 - q)x^{(2 m+1)} (y/1-x)^m + 1-x-xy/1-x (k_m - qx^my/1-x k_m-1). where k_m = ∑_æ=0 ^m-1 x^mj-(2j) (y/1 - x)^j. Here x and y respectively track the composition size and number of parts, whilst q tracks the number of such staircases contained.

Original language	English
Journal	Online Journal of Analytic Combinatorics
Issue number	11
State	Published - 2016

Bibliographical note

Publisher Copyright:
© 2016, Department of Computer Science. All rights reserved.

Keywords

Composition
Generating function

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

Cite this

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title = "Counting staircases in integer compositions",

abstract = "The main theorem establishes the generating function F which counts the number of times the staircase 1+2+3+⋯m+ fits inside an integer composition of n. F = km - qxmy/1-x km-1/(1 - q)x(2 m+1) (y/1-x)m + 1-x-xy/1-x (km - qxmy/1-x km-1). where km = ∑{\ae}=0 m-1 xmj-(2j) (y/1 - x)j. Here x and y respectively track the composition size and number of parts, whilst q tracks the number of such staircases contained.",

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author = "Blecher Aubrey and Mansour Toufik",

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journal = "Online Journal of Analytic Combinatorics",

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publisher = "Department of Computer Science, University of Auckland",

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AU - Toufik, Mansour

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AB - The main theorem establishes the generating function F which counts the number of times the staircase 1+2+3+⋯m+ fits inside an integer composition of n. F = km - qxmy/1-x km-1/(1 - q)x(2 m+1) (y/1-x)m + 1-x-xy/1-x (km - qxmy/1-x km-1). where km = ∑æ=0 m-1 xmj-(2j) (y/1 - x)j. Here x and y respectively track the composition size and number of parts, whilst q tracks the number of such staircases contained.

KW - Composition

KW - Generating function

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SN - 1931-3365

JO - Online Journal of Analytic Combinatorics

JF - Online Journal of Analytic Combinatorics

IS - 11

ER -

Counting staircases in integer compositions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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Cite this