A276724 -id:A276724 - 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).

A276842 Number of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most eight elements.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 181440, 756000, 3160080, 13602600, 60926580, 285159966, 1396646055, 7158444465, 35019420060, 165994449732, 774542703708, 3596199290264, 16729008975792, 78297230499296, 369560004222048, 1760728654735744, 8392003472443920

Offset: 0

Alois P. Heinz, Sep 20 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, G.f. for A276842

Column k=8 of A276837.

Cf. A276724.

G.f.: see link above.

A276898 Number of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most eight elements.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 6541426, 82575992, 1105638712, 15740668640, 238286993344, 3830996668880, 65290928665336, 1177060804189460, 22361980927148976, 446348461695232608, 9336784271669066016, 204240575957464571232, 4663193421334759826472

Offset: 0

Alois P. Heinz, Sep 21 2016

nonn

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

Column k=8 of A276890.

Cf. A276724.

a(n) ~ exp(7) * n!. - Vaclav Kotesovec, Sep 22 2016

A320558 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most eight elements and for at least one block c the smallest integer interval containing c has exactly eight elements.

Original entry on oeis.org

877, 5852, 27267, 110545, 422396, 1578192, 5877165, 22355618, 87597223, 345223398, 1352883364, 5249340393, 20158426185, 76729396494, 290259302392, 1094289866107, 4121529511428, 15518374075986, 58402401729381, 219602989556557, 824720185307142, 3092742982300231

Offset: 8

Alois P. Heinz, Oct 15 2018

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 of A276727.

Cf. A276723, A276724, A320621.

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)))(8):
seq(a(n), n=8..50);

a(n) = A276724(n) - A276723(n).

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).

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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A276842 Number of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most eight elements.

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A276898 Number of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most eight elements.

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A320558 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most eight elements and for at least one block c the smallest integer interval containing c has exactly eight 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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