A263757 -id:A263757 - cp's OEIS frontend

A276727 Number T(n,k) of set partitions of [n] where k is minimal such that for each block b the smallest integer interval containing b has at most k elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 4, 5, 5, 0, 1, 7, 12, 17, 15, 0, 1, 12, 29, 45, 64, 52, 0, 1, 20, 66, 121, 201, 265, 203, 0, 1, 33, 145, 336, 585, 966, 1197, 877, 0, 1, 54, 315, 901, 1741, 3172, 4971, 5852, 4140, 0, 1, 88, 676, 2347, 5375, 10100, 18223, 27267, 30751, 21147

Offset: 0

Alois P. Heinz, Sep 16 2016

			T(4,1) = 1: 1|2|3|4.
T(4,2) = 4: 12|34, 12|3|4, 1|23|4, 1|2|34.
T(4,3) = 5: 123|4, 13|24, 13|2|4, 1|234, 1|24|3.
T(4,4) = 5: 1234, 124|3, 134|2, 14|23, 14|2|3.
T(5,4) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   2;
  0, 1,  4,   5,   5;
  0, 1,  7,  12,  17,  15;
  0, 1, 12,  29,  45,  64,  52;
  0, 1, 20,  66, 121, 201, 265,  203;
  0, 1, 33, 145, 336, 585, 966, 1197, 877;
  ...

Alois P. Heinz, Rows n = 0..20, flattened

Columns k=0-10 give: A000007, A057427, A000071(n+1), A320553, A320554, A320555, A320556, A320557, A320558, A320559, A320560.
Row sums give A000110.
Main diagonal gives A000110(n-1) for n>0.
T(2n,n) gives A276728.
Cf. A263757, A276719, A276891.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k - 1]]]]; T [n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 12}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)

T(n,k) = A276719(n,k) - A276719(n,k-1) for k>0, T(n,0) = A000007(n).

A276837 Number A(n,k) of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 6, 5, 1, 0, 1, 1, 2, 6, 12, 8, 1, 0, 1, 1, 2, 6, 24, 25, 13, 1, 0, 1, 1, 2, 6, 24, 60, 57, 21, 1, 0, 1, 1, 2, 6, 24, 120, 150, 124, 34, 1, 0, 1, 1, 2, 6, 24, 120, 360, 399, 268, 55, 1, 0

Offset: 0

Alois P. Heinz, Sep 20 2016

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,    1,     1,     1, ...
  0, 1,  1,   1,    1,    1,    1,     1,     1, ...
  0, 1,  2,   2,    2,    2,    2,     2,     2, ...
  0, 1,  3,   6,    6,    6,    6,     6,     6, ...
  0, 1,  5,  12,   24,   24,   24,    24,    24, ...
  0, 1,  8,  25,   60,  120,  120,   120,   120, ...
  0, 1, 13,  57,  150,  360,  720,   720,   720, ...
  0, 1, 21, 124,  399, 1050, 2520,  5040,  5040, ...
  0, 1, 34, 268, 1145, 3192, 8400, 20160, 40320, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019

Columns k=0-10 give: A000007, A000012, A000045(n+1), A214663, A276838, A276839, A276840, A276841, A276842, A276843, A276844.
Main diagonal gives A000142.
Cf. A263757, A276719.

A(n,k+1) - A(n,k) = A263757(n,k) for n>0.

A276974 Number T(n,k) of permutations of [n] where the minimal distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 1, 1, 0, 19, 3, 1, 1, 0, 103, 12, 3, 1, 1, 0, 651, 54, 10, 3, 1, 1, 0, 4702, 281, 42, 10, 3, 1, 1, 0, 38413, 1652, 203, 37, 10, 3, 1, 1, 0, 350559, 11017, 1086, 166, 37, 10, 3, 1, 1, 0, 3539511, 81665, 6564, 857, 151, 37, 10, 3, 1, 1, 0, 39196758, 669948, 44265, 4900, 726, 151, 37, 10, 3, 1, 1

Offset: 0

Alois P. Heinz, Sep 23 2016

			T(3,1) = 4: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3).
T(3,2) = 1: (1,3)(2).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
  1;
  0,       1;
  0,       1,     1;
  0,       4,     1,    1;
  0,      19,     3,    1,   1;
  0,     103,    12,    3,   1,   1;
  0,     651,    54,   10,   3,   1,  1;
  0,    4702,   281,   42,  10,   3,  1,  1;
  0,   38413,  1652,  203,  37,  10,  3,  1, 1;
  0,  350559, 11017, 1086, 166,  37, 10,  3, 1, 1;
  0, 3539511, 81665, 6564, 857, 151, 37, 10, 3, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..12, flattened
Per Alexandersson et al., d-regular partitions and permutations, MathOverflow, 2014

Columns k=0-1 give: A000007, A276975.
Row sums give A000142.
T(2n,n) = A138378(n) = A005493(n-1) for n>0.
Cf. A239145, A263757, A277031.

A277031 Number T(n,k) of permutations of [n] where the minimal cyclic distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 0, 1, 0, 20, 3, 0, 1, 0, 109, 10, 0, 0, 1, 0, 668, 44, 7, 0, 0, 1, 0, 4801, 210, 28, 0, 0, 0, 1, 0, 38894, 1320, 90, 15, 0, 0, 0, 1, 0, 353811, 8439, 554, 75, 0, 0, 0, 0, 1, 0, 3561512, 63404, 3542, 310, 31, 0, 0, 0, 0, 1, 0, 39374609, 517418, 23298, 1276, 198, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Sep 25 2016

			T(3,1) = 5: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3), (1,3)(2).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
  1;
  0,       1;
  0,       1,     1;
  0,       5,     0,    1;
  0,      20,     3,    0,   1;
  0,     109,    10,    0,   0,  1;
  0,     668,    44,    7,   0,  0, 1;
  0,    4801,   210,   28,   0,  0, 0, 1;
  0,   38894,  1320,   90,  15,  0, 0, 0, 1;
  0,  353811,  8439,  554,  75,  0, 0, 0, 0, 1;
  0, 3561512, 63404, 3542, 310, 31, 0, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..12, flattened
Per Alexandersson et al., d-regular partitions and permutations, MathOverflow, 2014

Columns k=0-1 give: A000007, A277032.
Row sums give A000142.
T(2n,n) = A255047(n) = A000225(n) for n>0.
Cf. A239145, A263757, A276974.

cp's OEIS Frontend

A276727 Number T(n,k) of set partitions of [n] where k is minimal such that for each block b the smallest integer interval containing b has at most k elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A276837 Number A(n,k) of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A276974 Number T(n,k) of permutations of [n] where the minimal distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A277031 Number T(n,k) of permutations of [n] where the minimal cyclic distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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