A089479 Triangle T(n,k) read by rows, where T(n,k) = number of times the permanent of a real n X n (0,1)-matrix takes the value k, for n >= 0, 0 <= k <= n!.

0, 1, 1, 1, 9, 6, 1, 265, 150, 69, 18, 9, 0, 1, 27713, 13032, 10800, 4992, 4254, 1440, 1536, 576, 648, 24, 288, 96, 48, 0, 72, 0, 0, 0, 16, 0, 0, 0, 0, 0, 1, 10363361, 3513720, 4339440, 2626800, 3015450, 1451400, 1872800, 962400, 1295700, 425400, 873000

Offset: 0

Author

Hugo Pfoertner, Nov 05 2003

Keywords

• nonn
• tabf

Comments

The last element of each row is 1, corresponding to the n X n "all 1" matrix with permanent = n!. The first 4 rows were provided by Wouter Meeussen. The 6th row was computed by Gordon F. Royle: 13906734081, 2722682160, 4513642920, 3177532800, 4466769300, 2396826720, 3710999520, 2065521600, 3253760550, 1468314000, 2641593600, 1350475200, 2210277600, 1034061120,... .

Examples

			Triangle begins:
    0,     1;
    1,     1;
    9,     6,     1;
  265,   150,    69,   18,    9,    0,    1;
27713, 13032, 10800, 4992, 4254, 1440, 1536, 576, 648, 24, 288,
                   96, 48, 0, 72, 0, 0, 0, 16, 0, 0, 0, 0, 0, 1;
  ...
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..159 (rows 0..5, flattened)

Crossrefs

T(n,0) = A088672(n), T(n,1) = A089482(n). The n-th row of the table contains A087983(n) nonzero entries. For n>2 A089477(n) gives the position of the first zero entry in the n-th row.

Cf. A089480 (occurrence counts for permanents of non-singular (0,1)-matrices), A089481 (occurrence counts for permanents of singular (0,1)-matrices).

Cf. A000290, A038507 (row lengths), A002416 (row sums).

Cf. A036781, A089478.

Formula

From Geoffrey Critzer, Dec 20 2023: (Start)

Sum_{k=1..n!} T(n,k) = A227414(n).

For n>2, T(n,n!-(n-1)!) = n^2, the number of matrices with exactly one 0 entry. (End)

Extensions

Edited by Alois P. Heinz, Dec 20 2023