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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.)
##### Keywords:
permanental bounds for $$(0,1)$$-matrices
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