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<p(i)]]; triangle T(n,k), n>=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

0,7

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.

Cf. A002467, A324563.

Sequence in context: A329637 A276834 A016684 * A276974 A122777 A103524

Adjacent sequences: A324561 A324562 A324563 * A324565 A324566 A324567

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 06 2019

Status

approved