A276726 -id:A276726 - 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).

A276844 Number of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most ten elements.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 19958400, 99792000, 492972480, 2475673200, 12794581800, 68438391840, 379887726960, 2190484006290, 13122638916147, 81648757479285, 485188719524460, 2787398328848820, 15702226260438924, 87625414510220472

Offset: 0

Alois P. Heinz, Sep 20 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..64

Column k=10 of A276837.

Cf. A276726.

A276900 Number of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most ten elements.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, 1520385010, 23523325304, 380789137336, 6467270373536, 115360990236256, 2161133759447744, 42490485277902688, 875847933546165968, 18903517993567133752, 426618313415065361204, 10043227358229156346032

Offset: 0

Alois P. Heinz, Sep 21 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=10 of A276890.

Cf. A276726.

a(n) ~ exp(9) * n!. - Vaclav Kotesovec, Sep 22 2016

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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A276844 Number of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most ten elements.

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A276900 Number of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most ten elements.

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Author

Keywords

Links

Crossrefs

Formula

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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Author

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Maple

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