×

Asymptotic enumeration of lonesum matrices. (English) Zbl 1457.05010

Summary: We provide bivariate asymptotics for the poly-Bernoulli numbers, a combinatorial array that enumerates lonesum matrices, using the methods of Analytic Combinatorics in Several Variables (ACSV). For the diagonal asymptotic (i.e., for the special case of square lonesum matrices) we present an alternative proof based on Parseval’s identity. In addition, we provide an application in Algebraic Statistics on the asymptotic ML-degree of the bivariate multinomial missing data problem, and strengthen an existing result on asymptotic enumeration of permutations having a specified excedance set.

MSC:

05A16 Asymptotic enumeration
05A15 Exact enumeration problems, generating functions
05A05 Permutations, words, matrices
62R01 Algebraic statistics
11B68 Bernoulli and Euler numbers and polynomials
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahlfors, L. V., Complex analysis, (An Introduction to the Theory of Analytic Functions of One Complex Variable. An Introduction to the Theory of Analytic Functions of One Complex Variable, International Series in Pure and Applied Mathematics (1978), McGraw-Hill Book Co.: McGraw-Hill Book Co. New York)
[2] Arakawa, T.; Kaneko, M., Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J., 153, 189-209 (1999) · Zbl 0932.11055
[3] Arakawa, T.; Kaneko, M., On poly-Bernoulli numbers, Comment. Math. Univ. St. Pauli, 48, 2, 159-167 (1999) · Zbl 0994.11009
[4] Baryshnikov, Y.; Pemantle, R., Asymptotics of multivariate sequences, part III: quadratic points, Adv. Math., 228, 6, 3127-3206 (2011) · Zbl 1252.05012
[5] Bényi, B.; Hajnal, P., Combinatorics of poly-Bernoulli numbers, Studia Sci. Math. Hung., 52, 4, 537-558 (2015) · Zbl 1374.05002
[6] Bényi, B.; Hajnal, P., Combinatorial properties of poly-Bernoulli relatives, Integers, 17, Article A31 pp. (2017) · Zbl 1412.11034
[7] Brewbaker, C., A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, Integers, 8, 1, Article A02 pp. (2008), electronic only · Zbl 1165.11022
[8] Cameron, P. J.; Glass, C. A.; Schumacher, R. U., Acyclic orientations and poly-Bernoulli numbers (2014), preprint
[9] de Andrade, R. F.; Lundberg, E.; Nagle, B., Asymptotics of the extremal excedance set statistic, Eur. J. Comb., 46, 75-88 (2015) · Zbl 1307.05003
[10] de Bruijn, N. G., Asymptotic Methods in Analysis (1981), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0556.41021
[11] Ehrenborg, R.; Steingrímsson, E., The excedance set of a permutation, Adv. Appl. Math., 24, 3, 284-299 (2000) · Zbl 0957.05006
[12] Fink, A.; Rajchgot, J.; Sullivant, S., Matrix Schubert varieties and Gaussian conditional independence models, J. Algebraic Comb., 44, 4, 1009-1046 (2016) · Zbl 1411.14060
[13] Hoşten, S.; Sullivant, S., The algebraic complexity of maximum likelihood estimation for bivariate missing data, (Algebraic and Geometric Methods in Statistics (2010), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 123-133
[14] Kaneko, M., Poly-Bernoulli numbers, J. Théor. Nr. Bordx., 9, 1, 221-228 (1997) · Zbl 0887.11011
[15] Letsou, W.; Cai, L., Noncommutative biology: sequential regulation of complex networks, PLoS Comput. Biol., 12, 1 (2016)
[16] Lovász, L.; Vesztergombi, K., Restricted permutations and Stirling numbers, (Combinatorics. Combinatorics, Proc. Fifth Hungarian Colloq., Keszthely, 1976, vol. II. Combinatorics. Combinatorics, Proc. Fifth Hungarian Colloq., Keszthely, 1976, vol. II, Colloq. Math. Soc. János Bolyai, vol. 18 (1978), North-Holland: North-Holland Amsterdam, New York), 731-738
[17] Melczer, S.; Mishna, M., Asymptotic lattice path enumeration using diagonals, Algorithmica, 75, 4, 782-811 (2016) · Zbl 1390.05015
[18] Melczer, S.; Wilson, M. C., Higher dimensional lattice walks: connecting combinatorial and analytic behavior, SIAM J. Discrete Math., 33, 4, 2140-2174 (2019) · Zbl 1433.05026
[19] Pemantle, R.; Wilson, M. C., Twenty combinatorial examples of asymptotics derived from multivariate generating functions, SIAM Rev., 50, 2, 199-272 (2008) · Zbl 1149.05003
[20] Pemantle, R.; Wilson, M. C., Analytic Combinatorics in Several Variables, Cambridge Studies in Advanced Mathematics, vol. 140 (2013), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1297.05004
[21] Ryser, H. J., Combinatorial properties of matrices of zeros and ones, Can. J. Math., 9, 371-377 (1957) · Zbl 0079.01102
[22] Stanley, R. P., Enumerative Combinatorics, vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49 (2012), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1247.05003
[23] Vesztergombi, K., Permutations with restriction of middle strength, Studia Sci. Math. Hung., 9, 181-185 (1974) · Zbl 0374.05004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.