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Combinatorial identities related to 2× 2 submatrices of recursive matrices
[article]

2018
*
arXiv
*
pre-print

We obtain some

arXiv:1808.05736v1
fatcat:4s4evywmw5adhhsj6tbgnbsk4m
*combinatorial**identities**related**to*these sums, which generalized the work*of*Sun and Ma in [ Electron. J. Combin. 2014] and [ European J. Combin. 2014]. ... We focus on the study*of*the sums*of**2*×*2*minors*of*certain*recursive**matrices*, the alternating sums*of*their*2*×*2*minors, and the sums*of*their*2*×*2*permanents. ... The second author is supported in part by the National Science Foundation*of*China (No. 11771330). ...##
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Combinatorics of poly-Bernoulli numbers

2015
*
Studia scientiarum mathematicarum Hungarica (Print)
*

Brewbaker was the first

doi:10.1556/012.2015.52.4.1325
fatcat:rnfsf5fw6zhnzc6gw4tljqw7a4
*to*give*combinatorial*interpretation*of*these numbers. He proved that B (−k) n counts the, so called, lonesum 0-1*matrices**of*size n × k. ... Our new interpretation, for example, gives a transparent,*combinatorial*explanation*of*Kaneko's*recursive*formula for poly-Bernoulli numbers. ... These*combinatorial*definitions give us the possibility*to*explain previous*identities*-originally proven by algebraic methods -*combinatorially*. ...##
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Restricted lonesum matrices
[article]

2018
*
arXiv
*
pre-print

These

arXiv:1711.10178v2
fatcat:wtdspsvqobfmddvpmmflledwp4
*matrices*are enumerated by the poly-Bernoulli numbers that are*related**to*the multiple zeta values and have a rich literature in number theory. ... Motivated*of*these facts, we study in this paper lonesum*matrices*with restriction on the number*of*columns and rows*of*the same type. ... Next we derive a*recursive**relation*, though this*recursion*is too complicated for practical use. It shows only the simple*recursive*structure*of*these*matrices*. Theorem 5. ...##
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On q-poly-Bernoulli numbers arising from combinatorial interpretations
[article]

2019
*
arXiv
*
pre-print

We also recall some

arXiv:1909.09949v1
fatcat:5qh3ezbdrba4nhohrbqjqsgttu
*relating*analytical results and ask for*combinatorial*interpretations. ... In this paper we present several natural q-analogues*of*the poly-Bernoulli numbers arising in*combinatorial*contexts. ... Is it possible*to*give a*combinatorial*interpretation*of*B (−k) n,q using Γ-free*matrices*or*related**combinatorial*objects? ...##
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Restricted lonesum matrices

2018
*
Annales Mathematicae et Informaticae
*

These

doi:10.33039/ami.2018.10.001
fatcat:p5l3dufivreg5iu4wirmlhzlwq
*matrices*are enumerated by the poly-Bernoulli numbers; a sequence*related**to*the multiple zeta values with a rich literature in number theory. ... Motivated*of*these facts, we study in this paper lonesum*matrices*with restriction on the number*of*columns and rows*of*the same type. ... Lonesum*matrices*have interesting*combinatorial*properties and they are in bijection with many other*combinatorial*objects. See [*2*, 3] for an actual list*of*the*related*objects. ...##
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Parallel Output-Sensitive Algorithms for Combinatorial and Linear Algebra Problems

2001
*
Journal of computer and system sciences (Print)
*

(

doi:10.1006/jcss.2000.1740
fatcat:zrtbk6gxz5h3xacnjcp3jhu6yy
*To*avoid unduly complicating the statement*of*our results, we dropped these more exact bounds from the abstract.) 399 PARALLEL AND OUTPUT-SENSITIVE ALGORITHMS ... Inputs are n_n*matrices*over a fixed ground field. Let P(n) and M(n) be the PRAM processor bounds for O(log n) time multiplication*of*two degree n polynomials, and n_n*matrices*, respectively. ... ACKNOWLEDGMENTS The problems and applications considered in this paper were proposed by Joseph Cheriyan, who made invaluable contributions*to*the presentation*of*these results. ...##
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Parallel and output sensitive algorithms for combinatorial and linear algebra problems

1993
*
Proceedings of the fifth annual ACM symposium on Parallel algorithms and architectures - SPAA '93
*

(

doi:10.1145/165231.165238
dblp:conf/spaa/CheriyanR93
fatcat:k7crlmgq6bhxfm3zj34q63bg3e
*To*avoid unduly complicating the statement*of*our results, we dropped these more exact bounds from the abstract.) 399 PARALLEL AND OUTPUT-SENSITIVE ALGORITHMS ... Inputs are n_n*matrices*over a fixed ground field. Let P(n) and M(n) be the PRAM processor bounds for O(log n) time multiplication*of*two degree n polynomials, and n_n*matrices*, respectively. ... ACKNOWLEDGMENTS The problems and applications considered in this paper were proposed by Joseph Cheriyan, who made invaluable contributions*to*the presentation*of*these results. ...##
###
Excluded permutation matrices and the Stanley?Wilf conjecture

2004
*
Journal of combinatorial theory. Series A
*

and a

doi:10.1016/s0097-3165(04)00051-2
fatcat:dww2s2hmwbdszjc5st6xtzmts4
*related*conjecture*of*Alon and Friedgut (J. ... Due*to*the work*of*Martin Klazar (D. Krob, A.A. Mikhalev, A.V. ... Among other things Martin Klazar showed us how our result gives the characterization stated in Corollary 10 and Miklo´s Bo´na showed us the first written reference*to*the growth rate*of*S n ðpÞ in [17 ...##
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Excluded permutation matrices and the Stanley–Wilf conjecture

2004
*
Journal of combinatorial theory. Series A
*

and a

doi:10.1016/j.jcta.2004.04.002
fatcat:fpfkuwfnufha7fu53kubfskbmi
*related*conjecture*of*Alon and Friedgut (J. ... Due*to*the work*of*Martin Klazar (D. Krob, A.A. Mikhalev, A.V. ... Among other things Martin Klazar showed us how our result gives the characterization stated in Corollary 10 and Miklo´s Bo´na showed us the first written reference*to*the growth rate*of*S n ðpÞ in [17 ...##
###
Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

2001
*
Electronic Journal of Combinatorics
*

The well - known

doi:10.37236/1595
fatcat:tblyjy6yjvczxkzdmc4znkouki
*relation**of*Hankel*matrices**to*orthogonal polynomials further yields a*combinatorial*application*of*the famous Berlekamp – Massey algorithm in Coding Theory, which can be applied in order ...*to*calculate the coefficients in the three – term recurrence*of*the family*of*orthogonal polynomials*related**to*the sequence*of*Hankel*matrices*. ... When D n is the*identity*matrix, then L n = M n and the matrix M n was used in [54]*to*derive*combinatorial**identities*as for Catalan -like numbers. ...##
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On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress
[article]

2018
*
arXiv
*
pre-print

Since a classic algorithm due

arXiv:1806.09189v1
fatcat:t34z2h65ybfkzgvaodxkucmdf4
*to*Freivalds verifies correctness*of*matrix products probabilistically in time O(n^*2*), our question is a relaxation*of*the open problem*of*derandomizing Freivalds' algorithm ... Motivated by studying the power*of*randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication*of*two n× n*matrices*can be performed ... on a draft*of*this paper. ...##
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Combinatorics of poly-Bernoulli numbers
[article]

2015
*
arXiv
*
pre-print

Brewbaker was the first

arXiv:1510.05765v1
fatcat:wy6ew4id5bexnovsmwlypbe7j4
*to*give*combinatorial*interpretation*of*these numbers. He proved that B_n^(-k) counts the so called lonesum 0-1*matrices**of*size n× k. ... Our new interpretation, for example, gives a transparent,*combinatorial*explanation*of*Kaneko's*recursive*formula for poly-Bernoulli numbers ... These*combinatorial*definitions give us the possibility*to*explain previous*identities*-originally proven by algebraic methods -*combinatorially*. ...##
###
Sometimes Travelling is Easy: The Master Tour Problem

1998
*
SIAM Journal on Discrete Mathematics
*

We deal with the problem

doi:10.1137/s0895480195281878
fatcat:z3j54k457fdafdadywdlhvnwfy
*of*deciding for a given instance*of*the TSP, whether there is a renumbering*of*the cities such that the corresponding renumbered distance matrix ful lls the Kalmanson conditions ... In 1975, Kalmanson proved that in case the distance matrix in the Travelling Salesman Problem (TSP) ful lls certain*combinatorial*conditions (nowadays called the Kalmanson conditions) then the TSP is solvable ... We would like*to*thank Bettina Klinz for a careful reading*of*the paper and many helpful comments. ...##
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Efficient Matrix Multiplication: The Sparse Power-of-2 Factorization
[article]

2020
*
arXiv
*
pre-print

The algorithm decomposes the given matrix into a product

arXiv:2002.04002v2
fatcat:axzpm6r2lfdbbco2fzx7zjzvfu
*of**matrices*whose entries are either zero or integer powers*of*two utilizing the principles*of*sparse recovery. ... We present an algorithm*to*reduce the computational effort for the multiplication*of*a given matrix with an unknown column vector. ...*To*approximate a 64 × 64 matrix, one would decompose it into 6*submatrices**of*size 6 × 64 and 4*submatrices**of*size 7×64. Then, each*of*the 10*submatrices*is approximated by (3) . ...##
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Permutation classes and polyomino classes with excluded submatrices

2015
*
Mathematical Structures in Computer Science
*

For both permutation classes and polyomino classes, we present an original way

doi:10.1017/s0960129515000250
fatcat:qcntcyu7i5dtnk2h6rwpmtkovy
*of*characterizing them by avoidance constraints (namely, with excluded*submatrices*) and we discuss how canonical such a description ... This article introduces an analogue*of*permutation classes in the context*of*polyominoes. ... We would like*to*thank Valentin Féray, Samanta Socci and Laurent Vuillon for helpful discussions on the topics*of*this paper. Part*of*...
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