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A comparison of two upper bounds on the permanent of \((0,1)\)-matrices. (Ein Vergleich zweier oberer Schranken für die Permanente von \((0,1)\)-Matrizen.) (German) Zbl 0885.15002
Summary: The permanental bounds for \((0,1)\)-matrices by Minc-Bregman (1973) and for fully indecomposable integral matrices by J. Donald, J. Elwin, R. Hager, and P. Salamon [Linear Algebra Appl. 61, 199-218 (1984; Zbl 0551.15006)] are, in general, not comparable, even if we restrict ourselves to the class of fully indecomposable \((0,1)\)-matrices \(A\). In this note we present sufficient conditions (in terms of the number of those row sums of \(A\) that equal 2) such that it is possible to predict immediately which of the two bounds yields the better estimation for a given \(A\). It is noteworthy that this can be done without computing the bounds explicitly. As an important tool we derive a couple of properties of a function that is closely related to the classical gamma function.
MSC:
15A15 Determinants, permanents, traces, other special matrix functions
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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