Combinatorics of poly-Bernoulli numbers

Beáta Bényi; Péter Hajnal

doi:10.1556/012.2015.52.4.1325

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Studia Scientiarum Mathematicarum Hungarica

Volume/Issue:: Volume 52: Issue 4

Combinatorics of poly-Bernoulli numbers

Authors:

Beáta Bényi

Beáta Bényi József Eötvös College, Bajcsy-Zsilinszky u. 14., Baja 6500, Hungarye-mail: benyi.beata@ejf.hu

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benyi.beata@ejf.hu

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Péter Hajnal

Péter Hajnal University of Szeged, Bolyai Institute, Aradi Vértanúk tere 1., Szeged 6720, Hungarye-mail: hajnal@math.u-szeged.hu

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hajnal@math.u-szeged.hu

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Pages:: 537–558

Online Publication Date:: Dec 2015

Publication Date:: 01 Dec 2015

Article Category:: Journal Article

DOI:: https://doi.org/10.1556/012.2015.52.4.1325

Keywords:: Primary 05A05; 05A19; Poly-Bernoulli numbers; enumeration; combinatorial methods; bijective proofs; excluded submatrices

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The B_n^(k) poly-Bernoulli numbers — a natural generalization of classical Bernoulli numbers (B_n = B_n⁽¹⁾) — were introduced by Kaneko in 1997. When the parameter k is negative then B_n^(k) is a nonnegative number. 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. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko’s recursive formula for poly-Bernoulli numbers.

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Studia Scientiarum Mathematicarum Hungarica

Print ISSN:: 0081-6906

Online ISSN:: 1588-2896

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András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics)

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Imre BÁRÁNY (Rényi Institute of Mathematics)
Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
Péter CSIKVÁRI (ELTE, Budapest)
Joshua GREENE (Boston College)
Penny HAXELL (University of Waterloo)
Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
Ron HOLZMAN (Technion, Haifa)
Satoru IWATA (University of Tokyo)
Tibor JORDÁN (ELTE, Budapest)
Roy MESHULAM (Technion, Haifa)
Frédéric MEUNIER (École des Ponts ParisTech)
Márton NASZÓDI (ELTE, Budapest)
Eran NEVO (Hebrew University of Jerusalem)
János PACH (Rényi Institute of Mathematics)
Péter Pál PACH (BME, Budapest)
Andrew SUK (University of California, San Diego)
Zoltán SZABÓ (Princeton University)
Martin TANCER (Charles University, Prague)
Gábor TARDOS (Rényi Institute of Mathematics)
Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
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