Enumeration Problems for a Linear Congruence Equation

Tian-Xiao He; Wun-Seng Chou; Peter J. Shiue

doi:10.11650/tjm.18.2014.2295

Enumeration Problems for a Linear Congruence Equation

Tian-Xiao He, Wun-Seng Chou, Peter J. Shiue

Illinois Wesleyan University

Research output: Journal Article › Article › peer-review

Abstract

<div class="line" id="line-5"> Let m ≥ 2 and r ≥ 1 be integers and let c &Jukcy; Zm = {0, 1, …,m &boxh; 1}. In this paper, we give an upper bound and a lower bound for the number of unordered solutions x1, …, xn &Jukcy; Zm of the congruence x1 + x2 + ••• + xr &equiv; c mod m. Exact formulae are also given when m or r is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has interrelationship with the restricted integer partition.</div>

Original language	American English
Journal	Taiwanese Journal of Mathematics
Volume	18
DOIs	https://doi.org/10.11650/tjm.18.2014.2295
State	Published - Feb 2014

Keywords

Catalan number
Congruence
generalized Catalan number
integer partition

Disciplines

Mathematics

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10.11650/tjm.18.2014.2295License: Unspecified

Cite this

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title = "Enumeration Problems for a Linear Congruence Equation",

abstract = " Let m ≥ 2 and r ≥ 1 be integers and let c &Jukcy; Zm = {0, 1, …,m &boxh; 1}. In this paper, we give an upper bound and a lower bound for the number of unordered solutions x1, …, xn &Jukcy; Zm of the congruence x1 + x2 + ••• + xr &equiv; c mod m. Exact formulae are also given when m or r is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has interrelationship with the restricted integer partition.",

keywords = "Catalan number, Congruence, generalized Catalan number, integer partition",

author = "Tian-Xiao He and Wun-Seng Chou and Shiue, {Peter J.}",

note = "Let $m \geq 2$ and $r \geq 1$ be integers and let $c \in Z_m = \{0, 1,\dots, m - 1\}$. In this paper, we give an upper bound and a lower bound for the number of unordered solutions $x_1, \dots, x_n \in Z_m$ of the congruence $x_1 + x_2 + \cdots + x_r \equiv c \mod m$.",

year = "2014",

month = feb,

doi = "10.11650/tjm.18.2014.2295",

language = "American English",

volume = "18",

journal = "Taiwanese Journal of Mathematics",

}

TY - JOUR

T1 - Enumeration Problems for a Linear Congruence Equation

AU - He, Tian-Xiao

AU - Chou, Wun-Seng

AU - Shiue, Peter J.

N1 - Let $m \geq 2$ and $r \geq 1$ be integers and let $c \in Z_m = \{0, 1,\dots, m - 1\}$. In this paper, we give an upper bound and a lower bound for the number of unordered solutions $x_1, \dots, x_n \in Z_m$ of the congruence $x_1 + x_2 + \cdots + x_r \equiv c \mod m$.

PY - 2014/2

Y1 - 2014/2

N2 - Let m ≥ 2 and r ≥ 1 be integers and let c &Jukcy; Zm = {0, 1, …,m &boxh; 1}. In this paper, we give an upper bound and a lower bound for the number of unordered solutions x1, …, xn &Jukcy; Zm of the congruence x1 + x2 + ••• + xr &equiv; c mod m. Exact formulae are also given when m or r is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has interrelationship with the restricted integer partition.

AB - Let m ≥ 2 and r ≥ 1 be integers and let c &Jukcy; Zm = {0, 1, …,m &boxh; 1}. In this paper, we give an upper bound and a lower bound for the number of unordered solutions x1, …, xn &Jukcy; Zm of the congruence x1 + x2 + ••• + xr &equiv; c mod m. Exact formulae are also given when m or r is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has interrelationship with the restricted integer partition.

KW - Catalan number

KW - Congruence

KW - generalized Catalan number

KW - integer partition

UR - https://projecteuclid.org/journals/taiwanese-journal-of-mathematics/volume-18/issue-1/ENUMERATION-PROBLEMS-FOR-A-LINEAR-CONGRUENCE-EQUATION/10.11650/tjm.18.2014.2295.full

U2 - 10.11650/tjm.18.2014.2295

DO - 10.11650/tjm.18.2014.2295

M3 - Article

VL - 18

JO - Taiwanese Journal of Mathematics

JF - Taiwanese Journal of Mathematics

ER -

Enumeration Problems for a Linear Congruence Equation

Abstract

Keywords

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