A290133 Number of unique X-rays of n X n binary matrices with exactly n ones.

1, 1, 2, 5, 8, 13, 21, 31, 45, 65, 92, 127, 175, 237, 318, 425, 561, 735, 959, 1241, 1597, 2047, 2607, 3305, 4174, 5247, 6569, 8197, 10189, 12621, 15588, 19189, 23551, 28829, 35189, 42841, 52033, 63039, 76197, 91903, 110603, 132831, 159215, 190463, 227416

Offset: 0

Author

Alois P. Heinz, Jul 20 2017

Keywords

• nonn

Comments

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

A unique X-ray allows reconstruction of the binary matrix.

Examples

			a(3) = 5: 00021, 00300, 02001, 10020, 12000.
a(4) = 8: 0000301, 0004000, 0030001, 0200020, 1000021, 1000300, 1030000, 1200001.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• 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

Cf. A290052, A290134.

Programs

Maple

b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
      `if`(n=m or n=0, 1, add(b(n-i*j, min(n-i*j, i-1))*
      `if`(j=1, 2, 1), j=0..min(2, n/i)))))(i*(i+1))
    end:
a:= n-> `if`(n=0, 1, 1+b(n, n-1)) :
seq(a(n), n=0..60);

Mathematica

b[n_, i_] := b[n, i] = Function [m, If[n > m, 0, If[n == m || n == 0, 1, Sum[b[n - i*j, Min[n - i*j, i - 1]]*If[j == 1, 2, 1], {j, 0, Min[2, n/i]} ]]]][i*(i + 1)];
a[n_] := If[n == 0, 1, 1 + b[n, n - 1]] ;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 06 2018