A330759 -id:A330759 - cp's OEIS frontend

A294617 Number of ways to choose a set partition of a strict integer partition of n.

1, 1, 1, 3, 3, 5, 10, 12, 17, 24, 44, 51, 76, 98, 138, 217, 272, 366, 493, 654, 848, 1284, 1560, 2115, 2718, 3610, 4550, 6024, 8230, 10296, 13354, 17144, 21926, 27903, 35556, 44644, 59959, 73456, 94109, 117735, 150078, 185800, 235719, 290818, 365334, 467923

Offset: 0

Gus Wiseman, Nov 05 2017

nonn

From Gus Wiseman, Sep 17 2024: (Start)
Also the number of strict integer compositions of n whose leaders, obtained by splitting into maximal increasing subsequences and taking the first term of each, are decreasing. For example, the strict composition (3,6,2,1,4) has maximal increasing subsequences ((3,6),(2),(1,4)), with leaders (3,2,1), so is counted under a(16). The a(0) = 1 through a(7) = 12 compositions are:
  ()  (1)  (2)  (3)    (4)    (5)    (6)      (7)
                (1,2)  (1,3)  (1,4)  (1,5)    (1,6)
                (2,1)  (3,1)  (2,3)  (2,4)    (2,5)
                              (3,2)  (4,2)    (3,4)
                              (4,1)  (5,1)    (4,3)
                                     (1,2,3)  (5,2)
                                     (2,1,3)  (6,1)
                                     (2,3,1)  (1,2,4)
                                     (3,1,2)  (2,1,4)
                                     (3,2,1)  (2,4,1)
                                              (4,1,2)
                                              (4,2,1)
(End)

			The a(6) = 10 set partitions are: {{6}}, {{1},{5}}, {{5,1}}, {{2},{4}}, {{4,2}}, {{1},{2},{3}}, {{1},{3,2}}, {{2,1},{3}}, {{3,1},{2}}, {{3,2,1}}.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Cf. A000009, A000110, A063834, A258466, A259936, A279375, A279785, A279790.
Row sums of A330460 and of A330759.
This is a strict case of A374689, weak version A189076.
A011782 counts compositions, strict A032020.
A238130, A238279, A333755 count compositions by number of runs.
Cf. A188920, A238343, A274174, A373949, A374518, A376263.
Cf. A374631, A374632, A374634, A374635, A374686, A374688.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, combinat[bell](t), b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, min(n-i, i-1), t+1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 07 2017

Table[Total[BellB[Length[#]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,25}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2, 0, If[n == 0, BellB[t], b[n, i - 1, t] + If[i > n, 0, b[n - i, Min[n - i, i - 1], t + 1]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A279375(n) <= a(n) <= A279790(n).

G.f.: Sum_{k>=0} Bell(k) * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020

A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 2, 0, 0, 0, 0, 4, 5, 1, 0, 0, 0, 0, 5, 6, 1, 0, 0, 0, 0, 0, 6, 9, 2, 0, 0, 0, 0, 0, 0, 8, 13, 3, 0, 0, 0, 0, 0, 0, 0, 10, 23, 10, 1, 0, 0, 0, 0, 0, 0, 0, 12, 27, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 15, 40, 19, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 18 2019

			Triangle begins:
  1
  0  1
  0  1  0
  0  2  1  0
  0  2  1  0  0
  0  3  2  0  0  0
  0  4  5  1  0  0  0
  0  5  6  1  0  0  0  0
  0  6  9  2  0  0  0  0  0
  0  8 13  3  0  0  0  0  0  0
  0 10 23 10  1  0  0  0  0  0  0
  0 12 27 11  1  0  0  0  0  0  0  0
  0 15 40 19  2  0  0  0  0  0  0  0  0
Row n = 8 counts the following set partitions:
  {{8}}      {{1},{7}}    {{1},{2},{5}}
  {{3,5}}    {{2},{6}}    {{1},{3},{4}}
  {{2,6}}    {{3},{5}}
  {{1,7}}    {{1},{3,4}}
  {{1,3,4}}  {{1},{2,5}}
  {{1,2,5}}  {{2},{1,5}}
             {{3},{1,4}}
             {{4},{1,3}}
             {{5},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n=0..50)

Row sums are A294617.
Column k = 1 is A000009 (n > 0).
Cf. A000110, A008277, A050342, A060016, A072706, A270995, A271619, A279375, A279785, A326701, A330459, A330462, A330463, A330759.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 k*
         b(n-i, t, k)+b(n-i, t, k+1))(min(n-i, i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..15);  # Alois P. Heinz, Dec 29 2019

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,2],Length[#]==k&&And[UnsameQ@@#,UnsameQ@@Join@@#]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, x^k, b[n, i-1, k] + Function[t, k*b[n-i, t, k] + b[n-i, t, k + 1]][Min[n-i, i-1]]]];
T[n_] := PadRight[CoefficientList[b[n, n, 0], x], n + 1];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n)))); vector(#v, n, my(p=v[n]); vector(n, k, sum(i=k, n, polcoef(p,i-1)*stirling(i-1, k-1, 2))))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

T(n,k) = Sum_{k <= i <= n} A060016(n,i) * A008277(i,k).

For n > 0, T(n,2) = Sum_{k = 1..n} (2^(k - 1) -1) * A060016(n,k).

A330765 Total number of blocks in all set partitions of strict integer partitions of n.

Original entry on oeis.org

0, 1, 1, 4, 4, 7, 17, 20, 30, 43, 90, 103, 160, 210, 304, 515, 646, 894, 1223, 1659, 2176, 3484, 4226, 5873, 7638, 10335, 13150, 17695, 24974, 31394, 41383, 53766, 69718, 89573, 115613, 146344, 201625, 247880, 322099, 406445, 524634, 654298, 839584, 1043012

Offset: 0

Alois P. Heinz, Dec 29 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Wikipedia, Partition of a set

Cf. A003056, A330460, A330759.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n-i, t, k)*k
        +b(n-i, t, k+1))(min(n-i, i-1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n==0, k, b[n, i-1, k] + b[n-i, #, k] k + b[n-i, #, k+1]&[Min[n-i, i-1]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, May 08 2020, after Maple *)

a(n) = Sum_{k=1..A003056(n)} k * A330460(n,k).

a(n) = Sum_{k=1..A003056(n)} k * A330759(n,k).

cp's OEIS Frontend

A294617 Number of ways to choose a set partition of a strict integer partition of n.

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A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

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A330765 Total number of blocks in all set partitions of strict integer partitions of n.

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Maple

Mathematica

Formula