A290057 Number T(n,k) of X-rays of n X n binary matrices with exactly k ones; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.

1, 1, 1, 1, 3, 4, 3, 1, 1, 5, 13, 23, 30, 30, 23, 13, 5, 1, 1, 7, 26, 68, 139, 234, 334, 411, 440, 411, 334, 234, 139, 68, 26, 7, 1, 1, 9, 43, 145, 386, 860, 1660, 2838, 4362, 6090, 7779, 9135, 9892, 9892, 9135, 7779, 6090, 4362, 2838, 1660, 860, 386, 145, 43, 9, 1

Offset: 0

Author

Alois P. Heinz, Jul 19 2017

Keywords

• nonn
• tabf

Comments

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

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 for k>n^2.

Examples

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 3,  4,  3,   1;
  1, 5, 13, 23,  30,  30,  23,  13,   5,   1;
  1, 7, 26, 68, 139, 234, 334, 411, 440, 411, 334, 234, 139, 68, 26, 7, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..40, flattened
• 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

Columns k=0-2 give: A000012, A004273, A091823(n-1) for n>1.

Main diagonal gives A290052.

Row sums give A010790.

Cf. A000290, A290058.

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:
T:= (n, k)-> b(k, n, 1):
seq(seq(T(n, k), k=0..n^2), n=0..7);

Mathematica

b[n_,i_,t_]:= b[n, i, t] = Function[{m, jm}, If[n>m, 0, If[n==m, 1, Sum[b[n-j, i-t, 1-t], {j, 0, jm}]]]][i*(i+1-t), Min[i, n]]; T[n_, k_]:= b[k, n, 1]; Table[T[n, k], {n, 0, 7}, {k, 0, n^2}] // Flatten (* Jean-François Alcover, Aug 09 2017, translated from Maple *)

Formula

T(n,floor(n^2/2)) = A290058(n).

T(n,k) = T(n,n^2-k).