Counting staircases in integer compositions

Blecher Aubrey; Mansour Toufik

Counting staircases in integer compositions

Blecher Aubrey, Mansour Toufik

Department of Mathematics

University of Haifa

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

Keywords

Composition
Generating function

All Science Journal Classification (ASJC) codes

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.",

keywords = "Composition, Generating function",

author = "Blecher Aubrey and Mansour Toufik",

year = "2016",

language = "American English",

journal = "Online Journal of Analytic Combinatorics",

issn = "1931-3365",

publisher = "Combinatorial Press",

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AU - Aubrey, Blecher

AU - Toufik, Mansour

PY - 2016

Y1 - 2016

N2 - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=85002262295&partnerID=8YFLogxK

M3 - Article

SN - 1931-3365

JO - Online Journal of Analytic Combinatorics

JF - Online Journal of Analytic Combinatorics

IS - 11

ER -

Counting staircases in integer compositions

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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