A324563 Number T(n,k) of permutations p of [n] such that 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, 0, 1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 1, 7, 15, 0, 1, 1, 11, 31, 76, 0, 1, 1, 18, 60, 185, 455, 0, 1, 1, 29, 113, 435, 1275, 3186, 0, 1, 1, 47, 215, 1001, 3473, 10095, 25487, 0, 1, 1, 76, 406, 2299, 9289, 31315, 90109, 229384, 0, 1, 1, 123, 763, 5320, 24610, 95747, 313227, 895169, 2293839

Offset

0,10

Comments

Mirror image of triangle A324564.

Or as array: Number A(n,k) of permutations p of [n+k] such that k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+(n+k)*[i<p(i)]]; square array A(n,k), n>=0, k>=0, read by antidiagonals upwards.

Links

Alois P. Heinz, Rows n = 0..23, flattened

Wikipedia, Integer intervals

Wikipedia, Iverson bracket

Wikipedia, Permanent (mathematics)

Wikipedia, Permutation

Wikipedia, Symmetric group

Formula

T(n,k) = |{ p in S_n : k = max_{i=1..n} (1+i-p(i)+n*[i<p(i)]) }| for n>0, T(0,0) = 1.

T(n,k) = A008305(n,k) - A008305(n,k-1) for k > 0, T(n,0) = A000007(n).

Example

T(4,1) = A(3,1) = 1: 1234.
T(4,2) = A(2,2) = 1: 4123.
T(4,3) = A(1,3) = 7: 1423, 1432, 3124, 3214, 3412, 4132, 4213.
T(4,4) = A(0,4) = 15: 1243, 1324, 1342, 2134, 2143, 2314, 2341, 2413, 2431, 3142, 3241, 3421, 4231, 4312, 4321.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1,  4;
  0, 1, 1,  7,  15;
  0, 1, 1, 11,  31,   76;
  0, 1, 1, 18,  60,  185,   455;
  0, 1, 1, 29, 113,  435,  1275,   3186;
  0, 1, 1, 47, 215, 1001,  3473,  10095, 25487;
  ...
Square array A(n,k) begins:
  1, 1, 1,  4,  15,   76,   455,   3186, ...
  0, 1, 1,  7,  31,  185,  1275,  10095, ...
  0, 1, 1, 11,  60,  435,  3473,  31315, ...
  0, 1, 1, 18, 113, 1001,  9289,  95747, ...
  0, 1, 1, 29, 215, 2299, 24610, 290203, ...
  0, 1, 1, 47, 406, 5320, 65209, 876865, ...
  ...

Maple

b:= proc(n, k) option remember; `if`(n=k, n!,
       LinearAlgebra[Permanent](Matrix(n, (i, j)->
      `if`(i<=j and j<k+i or n+j<k+i, 1, 0))))
    end:
# as triangle:
T:= (n, k)-> b(n, k)-`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);
# as array:
A:= (n, k)-> b(n+k, k)-`if`(k=0, 0, 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[n == k, n!, Permanent[Table[If[i <= j && j < k + i || n + j < k + i, 1, 0], {i, 1, n}, {j, 1, n}]]];
(* as triangle: *)
T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k - 1]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
(* as array: *)
A[n_, k_] := b[n + k, k] - If[k == 0, 0, b[n + k, k - 1]]; Table[A[d - k, k], {d, 0, 10}, {k, 0, d}] // Flatten (* Jean-François Alcover, May 08 2019, after Alois P. Heinz *)

Crossrefs

Columns k=0, (1+2), 3-10 give: A000007, A000012, A000032 (for n>=3), A324631, A324632, A324633, A324634, A324635, A324636, A324637.

Diagonals of the triangle (rows of the array) n=0-10 give: A002467 (for k>0), A324621, A324622, A324623, A324624, A324625, A324626, A324627, A324628, A324629, A324630.

Row sums give A000142.

T(2n,n) or A(n,n) gives A324638.

Cf. A008305, A324564.

Sequence in context: A378833 A293301 A218453 * A186372 A200893 A294583

Adjacent sequences: A324560 A324561 A324562 * A324564 A324565 A324566

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 06 2019

Status

approved