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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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=0, 0<=k<=n, read by rows.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Mar 06 2019

Keywords

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=0, k>=0, read by antidiagonals upwards.

Examples

			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, ...
  ...

Links

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.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=k, n!,
       LinearAlgebra[Permanent](Matrix(n, (i, j)->
      `if`(i<=j and j 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 *)

Formula

T(n,k) = |{ p in S_n : k = max_{i=1..n} (1+i-p(i)+n*[i0, T(0,0) = 1.
T(n,k) = A008305(n,k) - A008305(n,k-1) for k > 0, T(n,0) = A000007(n).

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.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Mar 06 2019

Keywords

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

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 *)
Showing 1-2 of 2 results.

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