The Holens-Đoković conjecture on permanents fails!
The paper disproves the Holens-Doković conjecture on permanents that for a doubly-stochastic matrix \(A\) of order \(n\) and \(\sigma_i(A)\), the sum of the order \(i\) subpermanents of \(A\), \(in\sigma_i(A)\geq (-i+1)^2\sigma_{i-1}(A)\) for \(i=1, 2, \dots, n\) [cf. \textit{F. Holens}, Two aspects of doubly stochastic matrices: permutation matrices and the minimum permanent function, Ph.D. Thesis, Univ. Manitoba (1964); \textit{D. Ž. Đoković}, Mat. Vesnik, N. Ser. 4(19), 272-276 (1967; Zbl 0153.35403)]. The paper also shows that there is a natural counterpart of this conjecture for a \((0, 1)\)-matrix with constant line sum \(k\), showing that this associated conjecture holds when one of the following is satisfied: (1) \(k\leq 2\); (2) \(k\geq n-2\); (3) \(i\leq\frac nk +1\); (4) \(i\leq 5\).
scientific article; zbMATH DE number 1331737
scientific article