#
The Holens-Dokovic conjecture on permanents fails

Let A be a doubly stochastic matrix of order n and
σ_{i}(A) the sum of the order i subpermanents of A. The
HD conjecture says in
σ_{i}(A)≥(n-i+1)^{2}σ_{i-1}(A) for each
i=1,2,...,n. There is a natural counterpart of this conjecture for
(0,1)-matrices with constant line sum k. We show this associated
conjecture holds when k≤2, k≥n-2, i≤n/k+1 or i≤5 but that
it fails in general because it (wrongly) asserts a polynomial bound on
the ratio of near perfect to perfect matchings in k-regular bipartite
graphs.
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