login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 20
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A096459 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 5 00:15 EDT 2021. Contains 346456 sequences. (Running on oeis4.)