A290052 Number of X-rays of n X n binary matrices with exactly n ones.

1, 1, 4, 23, 139, 860, 5393, 34142, 217717, 1396346, 8997695, 58205686, 377775385, 2458841504, 16043226825, 104901986083, 687221188145, 4509605878736, 29636894936761, 195035340954186, 1285062484293880, 8476508261617168, 55969236979211755, 369900194873712830

Offset: 0

Author

Alois P. Heinz, Jul 19 2017

Keywords

• nonn

Comments

The X-ray of a matrix is defined as the sequence of antidiagonal sums.

Examples

			a(0) = 1: [].
a(1) = 1: 1.
a(2) = 4: 011, 020, 101, 110.
a(3) = 23: 00021, 00111, 00120, 00201, 00210, 00300, 01011, 01020, 01101, 01110, 01200, 02001, 02010, 02100, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
• Index entries for sequences related to binary matrices

Crossrefs

Main diagonal of A290057.

Cf. A019589, A014062, A290133.

Programs

Maple

b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      add(b(n-j, i-t, 1-t), j=0..min(i, n)))))(i*(i+1-t))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..30);

Mathematica

b[n_, i_, t_] := b[n, i, t] = Function[m, If[n > m, 0, If[n == m, 1, Sum[b[n - j, i - t, 1 - t], {j, 0, Min[i, n]}]]]][i*(i + 1 - t)];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

Formula

A019589(n) <= a(n) <= A014062(n).

a(n) ~ c * 3^(3*n) / (2^(2*n) * sqrt(n)), where c = 0.153294749730773567280925277269616968259180871352428154276351832424636097919... - Vaclav Kotesovec, Jul 22 2017