A324564 -id:A324564 - cp's OEIS frontend

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.

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

Alois P. Heinz, Mar 06 2019

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.

			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, ...
  ...

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

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.

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);

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 *)

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).

A324621 Number of permutations p of [1+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(1+n)*[i

Original entry on oeis.org

0, 1, 1, 7, 31, 185, 1275, 10095, 90109, 895169, 9793829, 116998199, 1515196619, 21143666585, 316260079951, 5047672782687, 85623656678457, 1538245254809537, 29176112648650441, 582614412521648359, 12217688610474042487, 268445509189890555577

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..449

Row n=1 of A324563 and column of A324564 (as array).

Cf. A000166, A000179.

a:= proc(n) option remember; `if`(n<5, [0, 1$2, 7, 31][n+1],
      ((2*n^4-3*n^3-2*n^2+n+4)*a(n-1) -(n^5-4*n^4+7*n^2+6*n-14)*
       a(n-2) -(n^5-2*n^4-4*n^3+2*n^2+13*n-12)*a(n-3)-(n-2)*
       (n^3+2*n^2+n-2)*a(n-4))/(n^3-n^2-2))
    end:
seq(a(n), n=0..23);

menage[n_] := If[n == 0, 1, 2n Sum[(-1)^k Binomial[2n-k, k] (n-k)!/(2n-k), {k, 0, n}]];
a[n_] := If[n == 0, 0, Subfactorial[n+1] - menage[n+1]];
a /@ Range[0, 21] (* Jean-François Alcover, Oct 28 2021 *)

a(n) = A000166(n+1) - A000179(n+1) for n < 0, a(0) = 0.

A324622 Number of permutations p of [2+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(2+n)*[i

Original entry on oeis.org

0, 1, 1, 11, 60, 435, 3473, 31315, 313227, 3445641, 41341502, 537313583, 7520316423, 112771887719, 1803821926465, 30656189582521, 551659191788556, 10478765887885181, 209522984620760153, 4398943767896801309, 96755196700729056267, 2224901906327124750355

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..448

Row n=2 of A324563 and column of A324564 (as array).

Cf. A000179, A000183.

a(n) = A000179(n+2) - A000183(n+2).

A324623 Number of permutations p of [3+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(3+n)*[i

Original entry on oeis.org

0, 1, 1, 18, 113, 1001, 9289, 95747, 1075779, 13129188, 173006731, 2449243815, 37082963875, 598045522873, 10236223969309, 185344819109346, 3539853769700281, 71122126197951465, 1499666213536206971, 33113352117542113491, 764116379880803291501

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..447

Row n=3 of A324563 and column of A324564 (as array).

Cf. A000183, A004307.

a(n) = A000183(n+3) - A004307(n+3).

A324624 Number of permutations p of [4+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(4+n)*[i

Original entry on oeis.org

0, 1, 1, 29, 215, 2299, 24610, 290203, 3664639, 49665695, 719356045, 11100719773, 181925519591, 3157018912485, 57848571473665, 1116400995778789, 22637359008083824, 481232567245746693, 10703530470036896333, 248615220921060645505, 6020095122314424497575

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Row n=4 of A324563 and column of A324564 (as array).

Cf. A004307, A189389.

a(n) = A004307(n+4) - A189389(n+4).

A324625 Number of permutations p of [5+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(5+n)*[i

Original entry on oeis.org

0, 1, 1, 47, 406, 5320, 65209, 876865, 12428079, 187013213, 2977639454, 50100075551, 889030153223, 16605705694513, 325842147818131, 6704025812865230, 144359437306938642, 3247712172059705741, 76210676599647821811, 1862449116865631232577

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Row n=5 of A324563 column of A324564 (as array).

Cf. A189389, A184965.

a(n) = A189389(n+5) - A184965(n+5).

A324626 Number of permutations p of [6+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(6+n)*[i

Original entry on oeis.org

0, 1, 1, 76, 763, 12277, 173111, 2653275, 42093128, 702648806, 12292640257, 225478889531, 4332332279039, 87108358452379, 1830631685045257, 40159929532511862, 918479788074790715, 21870877993310401595

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Row n=6 of A324563 and column of A324564 (as array).

A324627 Number of permutations p of [7+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(7+n)*[i

Original entry on oeis.org

0, 1, 1, 123, 1431, 28337, 457965, 8041166, 142688017, 2639058561, 50689730519, 1013206417391, 21074302033903, 456084254291809, 10265228833915967, 240124807728408475, 5833187694753202363

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Row n=7 of A324563 and column of A324564 (as array).

A324628 Number of permutations p of [8+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(8+n)*[i

Original entry on oeis.org

0, 1, 1, 199, 2676, 65469, 1209129, 24330731, 483906047, 9919457739, 209006411538, 4550241332823, 102418431583071, 2385231119694273, 57488444625865101, 1433825413465384003, 36995367724604395132

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Row n=8 of A324563 and column of A324564 (as array).

A324629 Number of permutations p of [9+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(9+n)*[i

Original entry on oeis.org

0, 1, 1, 322, 4993, 151401, 3188384, 73575295, 1638376021, 37296199680, 862203161593, 20436051653561, 497587696258955, 12467167386012869, 321709005522260383, 8554048660846031618

Offset: 0

Alois P. Heinz, Mar 09 2019

nonn

Row n=9 of A324563 and column of A324564 (as array).

cp's OEIS Frontend

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.

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A324621 Number of permutations p of [1+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(1+n)*[i

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A324622 Number of permutations p of [2+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(2+n)*[i

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A324623 Number of permutations p of [3+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(3+n)*[i

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A324624 Number of permutations p of [4+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(4+n)*[i

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A324625 Number of permutations p of [5+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(5+n)*[i

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A324626 Number of permutations p of [6+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(6+n)*[i

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A324627 Number of permutations p of [7+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(7+n)*[i

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A324628 Number of permutations p of [8+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(8+n)*[i

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A324629 Number of permutations p of [9+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(9+n)*[i

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