OFFSET
0,7
COMMENTS
Mirror image of A324563.
LINKS
Alois P. Heinz, Rows n = 0..23, flattened
Wikipedia, Integer intervals
Wikipedia, Iverson bracket
Wikipedia, Permanent (mathematics)
Wikipedia, Permutation
Wikipedia, Symmetric group
EXAMPLE
Triangle T(n,k) begins:
1;
1, 0;
1, 1, 0;
4, 1, 1, 0;
15, 7, 1, 1, 0;
76, 31, 11, 1, 1, 0;
455, 185, 60, 18, 1, 1, 0;
3186, 1275, 435, 113, 29, 1, 1, 0;
25487, 10095, 3473, 1001, 215, 47, 1, 1, 0;
...
Square array A(n,k) begins:
1, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
4, 7, 11, 18, 29, 47, ...
15, 31, 60, 113, 215, 406, ...
76, 185, 435, 1001, 2299, 5320, ...
455, 1275, 3473, 9289, 24610, 65209, ...
3186, 10095, 31315, 95747, 290203, 876865, ...
...
MAPLE
b:= proc(n, k) option remember; `if`(k>n, 0, `if`(k=0, n!,
LinearAlgebra[Permanent](Matrix(n, (i, j)->
`if`(j>=i and k+j<n+i or i>k+j, 1, 0)))))
end:
# as triangle:
T:= (n, k)-> b(n, k)-b(n, k+1):
seq(seq(T(n, k), k=0..n), n=0..10);
# as array:
A:= (n, k)-> b(n+k, k)-b(n+k, k+1):
seq(seq(A(d-k, k), k=0..d), d=0..10);
MATHEMATICA
b[n_, k_] := b[n, k] = If[k > n, 0, If[k == 0, n!, Permanent[Table[If[j >= i && k+j < n+i || i > k+j, 1, 0], {i, n}, {j, n}]]]];
(* as triangle: *)
T[n_, k_] := b[n, k] - b[n, k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
(* as array: *)
A[n_, k_] := b[n+k, k] - b[n+k, k+1];
Table[A[d-k, k], {d, 0, 10}, {k, 0, d}] // Flatten (* Jean-François Alcover, May 09 2019, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A002467 (for n>0), A324621, A324622, A324623, A324624, A324625, A324626, A324627, A324628, A324629, A324630.
Diagonals of the triangle (rows of the array) n=0, (1+2), 3-10 give: A000007, A000012, A000032 (for n>=3), A324631, A324632, A324633, A324634, A324635, A324636, A324637.
Row sums give A000142.
T(2n,n) or A(n,n) gives A324638.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 06 2019
STATUS
approved