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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.
3
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
0,5
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.
LINKS
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
FORMULA
T(n,floor(n^2/2)) = A290058(n).
T(n,k) = T(n,n^2-k).
EXAMPLE
T(2,2) = 4: 011, 020, 101, 110.
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;
...
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 *)
CROSSREFS
Columns k=0-2 give: A000012, A004273, A091823(n-1) for n>1.
Main diagonal gives A290052.
Row sums give A010790.
Sequence in context: A201162 A096646 A306234 * A384117 A384118 A249790
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jul 19 2017
STATUS
approved