An Inequality for Permanents of (0,1)-Matrices*

Let A be an n-square (0, 1)-matrix, let ri denote the i-th row sum of A, i = 1 ..... n, and let per(A) denote the permanent of A. Then per(A) ~< H ri q~/-2,.1 I + V T where equality can occur if and only if there exist permutation matrices P and Q such that PAQ is a direct sum of l-square and 2-square matrices all of whose entries are 1. I f A = (ai~) is an n-square mat r ix then the permanent o f A is defined by pe r (A) = ~ f i aio(i). (1) a e S n i=1 A n upto -da te survey o f the theory o f pe rmanen t s was given in [2]. M a n y proper t ies o f the p e r m a n e n t func t ion are s imilar to those of the de terminant . In par t icu la r , there is the fo l lowing ana log o f Lap lace expans ion by the k th c o l u m n pe r (A) = ~ aik per(A(i[k)), (2) * This work was supported by the Air Force Office of Scientific Research under Grant AFOSR 432-63.