M. Mays | West Virginia University (original) (raw)
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The idea of graph compositions generalizes both ordinary compositions of positive inte- gers and ... more The idea of graph compositions generalizes both ordinary compositions of positive inte- gers and partitions of flnite sets. In this paper we develop formulas, generating functions, and recurrence relations for composition counting functions for several families of graphs.
Journal of Combinatorial Theory, Series B, 2003
A double latin square of order 2n on symbols s 1 ; y; s n is a 2n  2n matrix A ¼ ða ij Þ in whic... more A double latin square of order 2n on symbols s 1 ; y; s n is a 2n  2n matrix A ¼ ða ij Þ in which each a ij is one of the symbols s 1 ; y; s n and each s k occurs twice in each row and twice in each column. For k ¼ 1; y; n let BðA; s k Þ be the bipartite graph with vertices r 1 ; y; r 2n ; c 1 ; y; c 2n and 4n edges ½r i ; c j corresponding to ordered pairs ði; jÞ such that a ij ¼ s k : We say that A is Hamiltonian if BðA; s k Þ is a cycle of length 4n for k ¼ 1; y; n: Two double latin squares ða ij Þ; ða 0 ij Þ of order 2n on symbols s 1 ; y; s n are said to be orthogonal if for each ordered pair ðs h ; s k Þ of symbols there are four ordered pairs ði; jÞ such that a ij ¼ s h ; a 0 ij ¼ s k :
Quaestiones Mathematicae, 2011
A composition of the positive integer n is a representation of n as an ordered sum of positive in... more A composition of the positive integer n is a representation of n as an ordered sum of positive integers n = a1 + a2 + · · · + am. It is well known that there are 2 n−1 compositions of n. An inversion in a composition is a pair of summands {ai, aj} for which i < j and ai > aj. The number of inversions of a composition is an indication of how far the composition is from a partition of n, which by convention uses a sequence of nondecreasing summands and has no inversions. We consider counting techniques for determining both the number of inversions in the set of compositions of n and the number of compositions of n with a given number of inversions. We provide explicit bijections to resolve several conjectures, and also consider asymptotic results. : 05A15, 05A16, 05A10, 05A17. 2. Inversion statistics: mean and variance. We begin with a generating function for counting inversions in the set of compositions. Let f (z, y, v) = ∑ σ∈C(n,m) z n y m v I(σ) , where C(n, m) is the set of compositions of n with m parts and I(σ) is the number of inversions in σ.
Integers
Graph compositions are related to compositions of positive integers and partitions of finite sets... more Graph compositions are related to compositions of positive integers and partitions of finite sets, and have applications in electrical networks. This paper provides extensions of a previously known result which states that
The idea of graph compositions generalizes both ordinary compositions of positive inte- gers and ... more The idea of graph compositions generalizes both ordinary compositions of positive inte- gers and partitions of flnite sets. In this paper we develop formulas, generating functions, and recurrence relations for composition counting functions for several families of graphs.
Journal of Combinatorial Theory, Series B, 2003
A double latin square of order 2n on symbols s 1 ; y; s n is a 2n  2n matrix A ¼ ða ij Þ in whic... more A double latin square of order 2n on symbols s 1 ; y; s n is a 2n  2n matrix A ¼ ða ij Þ in which each a ij is one of the symbols s 1 ; y; s n and each s k occurs twice in each row and twice in each column. For k ¼ 1; y; n let BðA; s k Þ be the bipartite graph with vertices r 1 ; y; r 2n ; c 1 ; y; c 2n and 4n edges ½r i ; c j corresponding to ordered pairs ði; jÞ such that a ij ¼ s k : We say that A is Hamiltonian if BðA; s k Þ is a cycle of length 4n for k ¼ 1; y; n: Two double latin squares ða ij Þ; ða 0 ij Þ of order 2n on symbols s 1 ; y; s n are said to be orthogonal if for each ordered pair ðs h ; s k Þ of symbols there are four ordered pairs ði; jÞ such that a ij ¼ s h ; a 0 ij ¼ s k :
Quaestiones Mathematicae, 2011
A composition of the positive integer n is a representation of n as an ordered sum of positive in... more A composition of the positive integer n is a representation of n as an ordered sum of positive integers n = a1 + a2 + · · · + am. It is well known that there are 2 n−1 compositions of n. An inversion in a composition is a pair of summands {ai, aj} for which i < j and ai > aj. The number of inversions of a composition is an indication of how far the composition is from a partition of n, which by convention uses a sequence of nondecreasing summands and has no inversions. We consider counting techniques for determining both the number of inversions in the set of compositions of n and the number of compositions of n with a given number of inversions. We provide explicit bijections to resolve several conjectures, and also consider asymptotic results. : 05A15, 05A16, 05A10, 05A17. 2. Inversion statistics: mean and variance. We begin with a generating function for counting inversions in the set of compositions. Let f (z, y, v) = ∑ σ∈C(n,m) z n y m v I(σ) , where C(n, m) is the set of compositions of n with m parts and I(σ) is the number of inversions in σ.
Integers
Graph compositions are related to compositions of positive integers and partitions of finite sets... more Graph compositions are related to compositions of positive integers and partitions of finite sets, and have applications in electrical networks. This paper provides extensions of a previously known result which states that