# Upper bounds on the permanent of dense 0-1 matrix

I am looking for some result useful in deriving following (conjecture?):

Let $A$ be an $n\times n$ matrix with $0-1$ entries. Suppose, that exactly $k\leqslant n$ entries are equal to $0$. Then $\mathrm{Per} A \leqslant n!\left(1-\frac{k}{2n}\right)$

Unfortunately, trying to derive this inequality from other estimates fails in general. For example, the inequality given in "A note on some upper bounds for permanents of (0,1)-matrices" failed with $k=4$.

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I should have naively guessed the upper bound $n!(1-k/n)$. Can you explain why the factor $1-\frac{k}{2n}$ is relevant? –  Denis Serre May 7 '11 at 20:35
The identity matrix $2\times 2$ with $k=2$ gives the equality $\mathrm{Per} I = 1 = 2!\left(1-\frac{2}{2\cdot 2}\right)$. –  Maciej S. May 7 '11 at 21:11
of course! I went to fast. –  Denis Serre May 8 '11 at 6:49

In fact, the maximum is achieved when no two of the 0's are in the same row or column, giving the maximum value $$S=\sum_{i=0}^k (-1)^i {k\choose i}(n-i)!.$$ Proof. Suppose that there are two 0's in the same row, say the first. Then some row, say the second, has all 1's. Expand the permanent by the first two rows. One of the $2\times 2$ submatrices is $\pmatrix{0&0\cr1&1\cr}$. Changing this to $\pmatrix{1&0\cr0&1\cr}$ does not decrease any of the $2\times 2$ submatrices from the first two rows, so the permanent of the entire matrix does not decrease. We can iterate this procedure (possibly using columns instead of rows) until no two 0's are in the same row and column, so the proof follows.

We now need to show that $S$ is at most $n!\left( 1 -\frac{k}{2n}\right)$. This is trivial for $k=1$, so assume $k>1$. By the Bonferroni inequalities (http://en.wikipedia.org/wiki/Boole's_inequality), $S$ is bounded by the first three terms, i.e., $$S\leq n!-k(n-1)!+{k\choose 2}(n-2)! = n!\left(1-\frac kn+\frac{{k\choose 2}}{n(n-1)}\right).$$ Since $k\leq n$ we have $$\frac{{k\choose 2}}{n(n-1)}\leq \frac{k}{2n},$$ and the result follows.

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\pmatrix{0&0\cr1&1\cr} gives $\pmatrix{0&0\cr1&1\cr}$ when you put a dollar sign at each end of it. –  Gerry Myerson May 8 '11 at 0:29
Thanks! I have revised my answer to use this notation. –  Richard Stanley May 8 '11 at 1:17