Vol. 95, No. 1, 1981

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Monotonicity of permanents of certain doubly stochastic matrices

David London

Vol. 95 (1981), No. 1, 125–131
Abstract

Let pk(A), k = 1,,n, denote the sum of the permanents of all k × k submatrices of the n × n matrix A.

We prove that

                  (         )
--n--  2n − k− 1
pk(In + Pn ) = n − k   k      ,  k = 1,⋅⋅⋅ ,n− 1,     (*)

where In and Pn are respectively the n × n identity matrix and the n × n permutation matrix with 1’s in positions (1,2),(2,3),,(n 1,n),(n,1). Using (), we prove that for n 3 and A = (In + Pn)2, the functions

pk((1 − 𝜃)Jn + 𝜃A),  k = 2,⋅⋅⋅ ,n,

are strictly monotonic increasing in the interval 0 𝜃 1. Here Jn is the n × n matrix all whose entries are equal to 1∕n.

Mathematical Subject Classification 2000
Primary: 15A15
Secondary: 15A51
Milestones
Received: 9 January 1980
Published: 1 July 1981
Authors
David London