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Commun. Korean Math. Soc. 2018; 33(4): 1055-1073

Online first article April 12, 2018      Printed October 31, 2018

https://doi.org/10.4134/CKMS.c170379

Copyright © The Korean Mathematical Society.

An alternative $q$-analogue of the Ruci\'nski-Voigt numbers

Wardah M. Bent-Usman, Amerah M. Dibagulun, Mahid M. Mangontarum, Charles B. Montero

Mindanao State University-Main Campus, Mindanao State University-Main Campus, Mindanao State University-Main Campus, Mindanao State University-Main Campus

Abstract

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In this paper, we define an alternative $q$-analogue of the Ruci\'nski-Voigt numbers. We obtain fundamental combinatorial properties such as recurrence relations, generating functions and explicit formulas which are shown to be $q$-deformations of similar properties for the Ruci\'nski-Voigt numbers, and are generalizations of the results obtained by other authors. A combinatorial interpretation in the context of $A$-tableaux is also given where convolution-type identities are consequently obtained. Lastly, we establish the matrix decompositions of the Ruci\'nski-Voigt and the $q$-Ruci\'nski-Voigt numbers.

Keywords: Stirling number, Ruci\'nski-Voigt number, Whitney number, $q$-analogue, $A$-tabluea, matrix decomposition

MSC numbers: Primary 11B83, 11B73, 05A30, 15B36

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