A276725 -id:A276725 - cp's OEIS frontend

A276719 Number A(n,k) of set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 5, 5, 1, 0, 1, 1, 2, 5, 10, 8, 1, 0, 1, 1, 2, 5, 15, 20, 13, 1, 0, 1, 1, 2, 5, 15, 37, 42, 21, 1, 0, 1, 1, 2, 5, 15, 52, 87, 87, 34, 1, 0, 1, 1, 2, 5, 15, 52, 151, 208, 179, 55, 1, 0, 1, 1, 2, 5, 15, 52, 203, 409, 515, 370, 89, 1, 0

Offset: 0

Alois P. Heinz, Sep 16 2016

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

			A(3,2) = 3: 12|3, 1|23, 1|2|3.
A(4,3) = 10: 123|4, 12|34, 12|3|4, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
A(5,4) = 37: 1234|5, 123|45, 123|4|5, 124|35, 124|3|5, 12|345, 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 1|2345, 1|234|5, 1|235|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|24|35, 1|24|3|5, 1|25|34, 1|2|345, 1|2|34|5, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
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,   5,   5,    5,    5,    5,    5, ...
  0, 1,  5,  10,  15,   15,   15,   15,   15, ...
  0, 1,  8,  20,  37,   52,   52,   52,   52, ...
  0, 1, 13,  42,  87,  151,  203,  203,  203, ...
  0, 1, 21,  87, 208,  409,  674,  877,  877, ...
  0, 1, 34, 179, 515, 1100, 2066, 3263, 4140, ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Wikipedia, Partition of a set

Columns k=0..10 give: A000007, A000012, A000045(n+1), A129847, A276720, A276721, A276722, A276723, A276724, A276725, A276726.
Main diagonal gives A000110.
A(n+1,n) gives A005493(n-1) for n>0.
Cf. A276727, A276837.

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)]))):
seq(seq(A(n, d-n), n=0..d), d=0..14);

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]]]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..k} A276727(n,i).

A276843 Number of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most nine elements.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 1814400, 8316000, 37920960, 176833800, 852972120, 4277399490, 22346336880, 121693555905, 690665206113, 3742590924108, 19625337285660, 101084160732660, 516806625700056, 2640952527095376, 13549247936670720

Offset: 0

Alois P. Heinz, Sep 20 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..128

Column k=9 of A276837.

Cf. A276725.

A276899 Number of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most nine elements.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 95160302, 1336605832, 19761235784, 308372439520, 5082485111648, 88427826212320, 1622190325391504, 31329432209039896, 635929197472661444, 13526250938401091568, 300743675140836904032, 6975365075051730713856

Offset: 0

Alois P. Heinz, Sep 21 2016

nonn

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

Column k=9 of A276890.

Cf. A276725.

a(n) ~ exp(8) * n!. - Vaclav Kotesovec, Sep 22 2016

A320559 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most nine elements and for at least one block c the smallest integer interval containing c has exactly nine elements.

Original entry on oeis.org

4140, 30751, 158766, 705926, 2928164, 11774145, 46852653, 186723275, 759062433, 3170429794, 13343960839, 56013146481, 233387096649, 963938933894, 3948441860748, 16062919807404, 65036845178255, 262641546675463, 1059920408695467, 4277149345637299

Offset: 9

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..512

Column k=9 of A276727.

Cf. A276724, A276725, A320622.

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)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(9):
seq(a(n), n=9..40);

a(n) = A276725(n) - A276724(n).

A320560 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most ten elements and for at least one block c the smallest integer interval containing c has exactly ten elements.

Original entry on oeis.org

21147, 172649, 977607, 4732307, 21196160, 91356135, 387221998, 1635077589, 6933701115, 29953216031, 132647186513, 592562183163, 2645622362009, 11748752847703, 51794376799161, 226629372792025, 984996790932516, 4257860514411454, 18336632254191876

Offset: 10

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..512

Column k=10 of A276727.

Cf. A276725, A276726, A320623.

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)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(10):
seq(a(n), n=10..40);

a(n) = A276726(n) - A276725(n).

cp's OEIS Frontend

A276719 Number A(n,k) of set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A276843 Number of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most nine elements.

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A276899 Number of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most nine elements.

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A320559 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most nine elements and for at least one block c the smallest integer interval containing c has exactly nine elements.

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A320560 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most ten elements and for at least one block c the smallest integer interval containing c has exactly ten elements.

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