A324564 Number T(n,k) of permutations p of [n] such that n-k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+n*[i=0, 0<=k<=n, read by rows.

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, 229384, 90109, 31315, 9289, 2299, 406, 76, 1, 1, 0, 2293839, 895169, 313227, 95747, 24610, 5320, 763, 123, 1, 1, 0

Offset: 0

Author

Alois P. Heinz, Mar 06 2019

Keywords

• nonn
• tabl

Comments

Mirror image of A324563.

Examples

			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, ...
   ...
		

Links

• Alois P. Heinz, Rows n = 0..23, flattened
• Wikipedia, Integer intervals
• Wikipedia, Iverson bracket
• Wikipedia, Permanent (mathematics)
• Wikipedia, Permutation
• Wikipedia, Symmetric group

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.

Cf. A002467, A324563.

Programs

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+jk+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 *)