Concerning the minimum of permanents on doubly stochastic circulants

Gerald Suchan

Vol. 95 (1981), No. 1, 213–217

DOI: 10.2140/pjm.1981.95.213

Abstract

Let P_n be the permutation matrix such that (P_n)_ij = 1 if j = i + 1 (mod n). Mine [2] proved that the minimum of the permanent on the collection of n × n doubly stochastic circulants αI_n + βP_n + γP_n² is in (1∕2ⁿ,1∕2ⁿ⁻¹], and if n ≧ 5 then the minimum is not achieved at (1∕3)I_n + (1∕3)P_n + (1∕3)P_n². This paper proves that if n ≧ 3 then the minimum of such permanents is less than 1∕2ⁿ⁻¹, and if n ∈{3,4} then this minimum is uniquely achieved at (1∕3)I_n + (1∕3)P_n + (1∕3)P_n².

Mathematical Subject Classification 2000

Primary: 15A15

Milestones

Received: 3 July 1973

Published: 1 July 1981