Abstract: A matrix is lonesum if it can be uniquely reconstructed from its row and column sums. Brewbaker computed the number of m×n binary lonesum matrices. Kaneko defined the poly-Bernoulli numbers of an integer index, and showed that the number of m×n binary lonesum matrices is equal to the mth poly-Bernoulli number of index -n. In this paper, we are interested in q-ary lonesum matrices. There are two types of lonesumness for q-ary matrices, namely strongly and weakly lonesum. We first study strongly lonesum matrices: We compute the number of m×n q-ary strongly lonesum matrices, and provide a generalization of Kaneko’s formulas by deriving the generating function for the number of m×n q-ary strongly lonesum matrices. Next, we study weakly lonesum matrices: We show that the number of forbidden patterns for q-ary weakly lonesum matrices is infinite if q⩾5, and construct some forbidden patterns for q=3,4. We also suggest an open problem related to ternary and quaternary weakly lonesum matrices.

Cite: Kim H.K. , Krotov D.S. , Lee J.Y.
Matrices uniquely determined by their lonesums
Linear Algebra and Its Applications. 2013. V.438. N7. P.3107-3123. DOI: 10.1016/j.laa.2012.11.027 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Nov 2, 2012
Accepted:	Nov 21, 2012
Published online:	Jan 5, 2013
Published print:	Apr 1, 2013

Identifiers:

Web of science:	WOS:000315830200017
Scopus:	2-s2.0-84873742671
Elibrary:	20433728
OpenAlex:	W1949430587

Citing:

DB	Citing
Web of science	4
Scopus	4
Elibrary	3
OpenAlex	10

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