A336130 -id:A336130 - cp's OEIS frontend

A336343 Number of ways to choose a strict partition of each part of a strict composition of n.

1, 1, 1, 4, 6, 11, 26, 39, 78, 142, 320, 488, 913, 1558, 2798, 5865, 9482, 16742, 28474, 50814, 82800, 172540, 266093, 472432, 790824, 1361460, 2251665, 3844412, 7205416, 11370048, 19483502, 32416924, 54367066, 88708832, 149179800, 239738369, 445689392

Offset: 0

Gus Wiseman, Jul 19 2020

nonn

A strict composition of n (A032020) is a finite sequence of distinct positive integers summing to n.

Is there a simple generating function?

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (2,1)    (3,1)      (3,2)
            (1),(2)  (1),(3)    (4,1)
            (2),(1)  (3),(1)    (1),(4)
                     (1),(2,1)  (2),(3)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1),(3,1)
                                (2,1),(2)
                                (2),(2,1)
                                (3,1),(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of strict partitions are A072706.
Set partitions of strict partitions are A294617.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A318683, A318684, A319794, A323583, A336128, A336130, A336132, A336133.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[strptn/@ctn],{ctn,Join@@Permutations/@strptn[n]}]],{n,0,10}]

\\ here Q(N) gives A000009 as a vector.
Q(n) = {Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)))}
seq(n)={my(b=Q(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*b[1+k] + O(x*x^n)))]} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=0} k! * [y^k](Product_{j>=1} 1 + y*x^j*A000009(j)). - Andrew Howroyd, Apr 16 2021

A336141 Number of ways to choose a strict composition of each part of an integer partition of n.

Original entry on oeis.org

1, 1, 2, 5, 9, 17, 41, 71, 138, 270, 518, 938, 1863, 3323, 6163, 11436, 20883, 37413, 69257, 122784, 221873, 397258, 708142, 1249955, 2236499, 3917628, 6909676, 12130972, 21251742, 36973609, 64788378, 112103360, 194628113, 336713377, 581527210, 1000153063

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(1) = 1 through a(5) = 17 ways:
  (1)  (2)      (3)          (4)              (5)
       (1),(1)  (1,2)        (1,3)            (1,4)
                (2,1)        (3,1)            (2,3)
                (2),(1)      (2),(2)          (3,2)
                (1),(1),(1)  (3),(1)          (4,1)
                             (1,2),(1)        (3),(2)
                             (2,1),(1)        (4),(1)
                             (2),(1),(1)      (1,2),(2)
                             (1),(1),(1),(1)  (1,3),(1)
                                              (2,1),(2)
                                              (3,1),(1)
                                              (2),(2),(1)
                                              (3),(1),(1)
                                              (1,2),(1),(1)
                                              (2,1),(1),(1)
                                              (2),(1),(1),(1)
                                              (1),(1),(1),(1),(1)

Alois P. Heinz, Table of n, a(n) for n = 0..5623

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 31 2020

Table[Length[Join@@Table[Tuples[Join@@Permutations/@Select[IntegerPartitions[#],UnsameQ@@#&]&/@ctn],{ctn,IntegerPartitions[n]}]],{n,0,10}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
     If[n==0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[n==0 || i==1, 1, g[n, i-1] +
     b[i, i, 0] g[n-i, Min[n-i, i]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

G.f.: Product_{k >= 1} 1/(1 - A032020(k)*x^k).

A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 17, 22, 26, 35, 40, 51, 60, 75, 86, 109, 124, 153, 175, 214, 243, 297, 336, 403, 456, 546, 614, 731, 821, 975, 1095, 1283, 1437, 1689, 1887, 2195, 2448, 2851, 3172, 3676, 4083, 4724, 5245, 6022, 6677, 7695, 8504, 9720

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(9) = 9 splittings:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)    (5,4)
                          (4,1)  (5,1)    (5,2)    (6,2)    (6,3)
                                 (3,2,1)  (6,1)    (7,1)    (7,2)
                                          (4,2,1)  (4,3,1)  (8,1)
                                                   (5,2,1)  (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)
                                                            (4),(3,2)
The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal sums is A318683.
The version with strictly decreasing sums is A318684.
The version with weakly decreasing sums is A319794.
The version with different sums is A336132.
Starting with a composition gives A304961.
Starting with a non-strict partition gives A336134.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336127, A336128, A336130, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 31, 40, 73, 98, 158, 204, 340, 420, 629, 819, 1202, 1494, 2174, 2665, 3759, 4688, 6349, 7806, 10788, 13035, 17244, 21128, 27750, 33499, 43941, 52627, 67957, 81773, 103658, 124047, 158628, 187788, 235162, 280188, 349612, 413120, 513952, 604568

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(5) = 15 splittings:
  (1)  (2)      (3)          (4)              (5)
       (1,1)    (2,1)        (2,2)            (3,2)
       (1),(1)  (1,1,1)      (3,1)            (4,1)
                (1),(1,1)    (2,1,1)          (2,2,1)
                (1),(1),(1)  (2),(2)          (3,1,1)
                             (1,1,1,1)        (2,1,1,1)
                             (2),(1,1)        (2),(2,1)
                             (1),(1,1,1)      (1,1,1,1,1)
                             (1,1),(1,1)      (2),(1,1,1)
                             (1),(1),(1,1)    (1),(1,1,1,1)
                             (1),(1),(1),(1)  (1,1),(1,1,1)
                                              (1),(1),(1,1,1)
                                              (1),(1,1),(1,1)
                                              (1),(1),(1),(1,1)
                                              (1),(1),(1),(1),(1)

Andrew Howroyd, Table of n, a(n) for n = 0..60

The version with weakly decreasing sums is A316245.
The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with strictly decreasing sums is A336135.
The version with different sums is A336131.
Starting with a composition gives A075900.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A304961, A305551, A318684, A323433, A336128, A336130, A336133.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],LessEqual@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r >= t && t >= s, self()(r,m,t,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m),s,t+m,1))); recurse(n,n,0,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 22, 41, 72, 142, 260, 454, 769, 1416, 2472, 4465, 7708, 13314, 23630, 40406, 68196, 119646, 203237, 343242, 586508, 993764, 1677187, 2824072, 4753066, 7934268, 13355658, 22229194, 36945828, 61555136, 102019156, 168474033, 279181966

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (1,2)    (1,3)      (1,4)
            (2,1)    (3,1)      (2,3)
            (2),(1)  (3),(1)    (3,2)
                     (1,2),(1)  (4,1)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1,2),(2)
                                (1,3),(1)
                                (2,1),(2)
                                (3,1),(1)

Alois P. Heinz, Table of n, a(n) for n = 0..7725

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 31 2020

strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[Join@@Permutations/@strptn[#]&/@ctn],{ctn,strptn[n]}]],{n,0,20}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
     If[n == 0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0,
     If[n == 0, 1, g[n, i-1] + b[i, i, 0]*g[n-i, Min[n-i, i-1]]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

G.f.: Product_{k >= 1} (1 + A032020(k)*x^k).

A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

Original entry on oeis.org

1, 2, 5, 14, 37, 98, 259, 682, 1791, 4697, 12303, 32196, 84199, 220087, 575067, 1502176, 3923117, 10244069, 26746171, 69825070, 182276806, 475804961, 1241965456, 3241732629, 8461261457, 22084402087, 57640875725, 150442742575, 392652788250, 1024810764496

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

By "distinct" we mean equal subsequences are counted only once. For example, the pair (1,1)(1) is counted only once even though (1) is a subsequence of (1,1) in two ways. The version with multiplicity is A025192.

			The a(3) = 14 pairings of a composition with a chosen subsequence:
  (3)()     (3)(3)
  (21)()    (21)(1)   (21)(2)    (21)(21)
  (12)()    (12)(1)   (12)(2)    (12)(12)
  (111)()   (111)(1)  (111)(11)  (111)(111)

Christian Sievers, Table of n, a(n) for n = 0..2000

For partitions we have A000712, composable A339006.
The homogeneous version is A011782, without containment A000302.
With multiplicity we have A025192, for partitions A070933.
The strict case is A032005.
The case of strict subsequences is A236002.
The composable case is A355384, homogeneous without containment A355388.
A075900 counts compositions of each part of a partition.
A304961 counts compositions of each part of a strict partition.
A307068 counts strict compositions of each part of a composition.
A336127 counts compositions of each part of a strict composition.
Cf. A011782, A022811, A032020, A063834, A133494, A181591, A323583, A331330, A336128, A336130, A336139, A355382, A355383.

Table[Sum[Length[Union[Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,6}]

lista(n)=my(f=sum(k=1,n,(x^k+x*O(x^n))/(1-x/(1-x)+x^k)));Vec((1-x)/((1-2*x)*(1-f))) \\ Christian Sievers, May 06 2025

G.f.: (1-x)/((1-2*x)*(1-f)) where f = Sum_{k>=1} x^k/(1-x/(1-x)+x^k) is the generating function for A331330. - Christian Sievers, May 06 2025

a(16) and beyond from Christian Sievers, May 06 2025

A336129 Number of strict compositions of divisors of n.

Original entry on oeis.org

1, 2, 4, 5, 6, 16, 14, 24, 31, 64, 66, 120, 134, 208, 360, 459, 618, 894, 1178, 1622, 2768, 3364, 4758, 6432, 8767, 11440, 15634, 24526, 30462, 42296, 55742, 75334, 98112, 131428, 168444, 258403, 315974, 432244, 558464, 753132, 958266, 1280840, 1621274

Offset: 1

Gus Wiseman, Jul 11 2020

nonn

A strict composition of k is a finite sequence of distinct positive integers summing to k.

			The a(1) = 1 through a(7) = 14 compositions:
  (1)  (1)  (1)    (1)    (1)    (1)      (1)
       (2)  (3)    (2)    (5)    (2)      (7)
            (1,2)  (4)    (1,4)  (3)      (1,6)
            (2,1)  (1,3)  (2,3)  (6)      (2,5)
                   (3,1)  (3,2)  (1,2)    (3,4)
                          (4,1)  (1,5)    (4,3)
                                 (2,1)    (5,2)
                                 (2,4)    (6,1)
                                 (4,2)    (1,2,4)
                                 (5,1)    (1,4,2)
                                 (1,2,3)  (2,1,4)
                                 (1,3,2)  (2,4,1)
                                 (2,1,3)  (4,1,2)
                                 (2,3,1)  (4,2,1)
                                 (3,1,2)
                                 (3,2,1)

Compositions of divisors are A034729.
Strict partitions of divisors are A047966.
Partitions of divisors are A047968.
Cf. A000005, A001970, A006951, A133494, A318683, A336128, A336130, A336132.

Table[Sum[Length[Join@@Permutations/@Select[IntegerPartitions[d],UnsameQ@@#&]],{d,Divisors[n]}],{n,12}]

Moebius transform is A032020 (strict compositions).

cp's OEIS Frontend

A336343 Number of ways to choose a strict partition of each part of a strict composition of n.

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A336141 Number of ways to choose a strict composition of each part of an integer partition of n.

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A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

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A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

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A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

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A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

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A336129 Number of strict compositions of divisors of n.

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