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

Fangfang Cai, Qing-Hu Hou, Yidong Sun, Arthur L.B. Yang
2018 arXiv   pre-print
We obtain some 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).  ...

### Combinatorics of poly-Bernoulli numbers

Beáta Bényi, Péter Hajnal
2015 Studia scientiarum mathematicarum Hungarica (Print)
Brewbaker was the first 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.  ...

### Restricted lonesum matrices [article]

Beata Benyi
2018 arXiv   pre-print
These 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.  ...

### On q-poly-Bernoulli numbers arising from combinatorial interpretations [article]

Beáta Bényi, José Luis Ramírez
2019 arXiv   pre-print
We also recall some 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?  ...

### Restricted lonesum matrices

Beáta Bényi
2018 Annales Mathematicae et Informaticae
These 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.  ...

### Parallel Output-Sensitive Algorithms for Combinatorial and Linear Algebra Problems

John H. Reif
2001 Journal of computer and system sciences (Print)
(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.  ...

### Parallel and output sensitive algorithms for combinatorial and linear algebra problems

Joseph Cheriyan, John H. Reif
1993 Proceedings of the fifth annual ACM symposium on Parallel algorithms and architectures - SPAA '93
(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

A MARCUS
2004 Journal of combinatorial theory. Series A
and a 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  ...

### Excluded permutation matrices and the Stanley–Wilf conjecture

Adam Marcus, Gábor Tardos
2004 Journal of combinatorial theory. Series A
and a 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

Ulrich Tamm
2001 Electronic Journal of Combinatorics
The well - known 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.  ...

### On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress [article]

Marvin Künnemann
2018 arXiv   pre-print
Since a classic algorithm due 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.  ...

### Combinatorics of poly-Bernoulli numbers [article]

Beáta Bényi, Peter Hajnal
2015 arXiv   pre-print
Brewbaker was the first 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

Vladimir G. Deuineko, Rüdiger Rudolf, Gerhard J. Woeginger
1998 SIAM Journal on Discrete Mathematics
We deal with the problem 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.  ...

### Efficient Matrix Multiplication: The Sparse Power-of-2 Factorization [article]

Ralf R. Müller, Bernhard Gäde, Ali Bereyhi
2020 arXiv   pre-print
The algorithm decomposes the given matrix into a product 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) .  ...

### Permutation classes and polyomino classes with excluded submatrices

DANIELA BATTAGLINO, MATHILDE BOUVEL, ANDREA FROSINI, SIMONE RINALDI
2015 Mathematical Structures in Computer Science
For both permutation classes and polyomino classes, we present an original way 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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