A279375 -id:A279375 - cp's OEIS frontend

A063834 Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.

1, 1, 3, 6, 15, 28, 66, 122, 266, 503, 1027, 1913, 3874, 7099, 13799, 25501, 48508, 88295, 165942, 299649, 554545, 997281, 1817984, 3245430, 5875438, 10410768, 18635587, 32885735, 58399350, 102381103, 180634057, 314957425, 551857780, 958031826, 1667918758

Offset: 0

Wouter Meeussen, Aug 21 2001

These are different from plane partitions.
For ordered partitions of partitions see A055887 which may be computed from A036036 and A048996. - Alford Arnold, May 19 2006
Twice partitioned numbers correspond to triangles (or compositions) in the multiorder of integer partitions. - Gus Wiseman, Oct 28 2015

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + 266*x^8 + ...
If n=6, a possible first partitioning is (3+3), resulting in the following second partitionings: ((3),(3)), ((3),(2+1)), ((3),(1+1+1)), ((2+1),(3)), ((2+1),(2+1)), ((2+1),(1+1+1)), ((1+1+1),(3)), ((1+1+1),(2+1)), ((1+1+1),(1+1+1)).

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n=1..500 from T. D. Noe)
Vaclav Kotesovec, Screenshot - A closed form formula for the constant c
Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Sequences enumerating triangles of integer partitions
Gus Wiseman, On the Categorical Structure of Weakly Ordered Sequences of Integer Partitions

Cf. A063835, A196545.
Cf. A036036, A048996, A055887.
Cf. A006906, A270995.
Cf. A007425, A047966, A047968, A271619, A279375, A279784-A279791.
The strict case is A296122.
Row sums of A321449.
Column k=2 of A323718.
Without singletons we have A327769, A358828, A358829.
For odd lengths we have A358823, A358824.
For distinct lengths we have A358830, A358912.
For strict partitions see A358914, A382524.
A000041 counts integer partitions, strict A000009.
A001970 counts multiset partitions of integer partitions.
Cf. A075900, A358831, A358833, A358906, A358908, A358909.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+`if`(i>n, 0, numbpart(i)*b(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 26 2015

Table[Plus @@ Apply[Times, IntegerPartitions[i] /. i_Integer :> PartitionsP[i], 2], {i, 36}]
(* second program: *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i > n, 0, PartitionsP[i]*b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 19 2016 */

G.f.: 1/Product_{k>0} (1-A000041(k)*x^k). n*a(n) = Sum_{k=1..n} b(k)*a(n-k), a(0) = 1, where b(k) = Sum_{d|k} d*A000041(d)^(k/d) = 1, 5, 10, 29, 36, 110, 106, ... . - Vladeta Jovovic, Jun 19 2003
From Vaclav Kotesovec, Mar 27 2016: (Start)
a(n) ~ c * 5^(n/4), where
c = 96146522937.7161898848278970039269600938032826... if n mod 4 = 0
c = 96146521894.9433858914667933636782092683849082... if n mod 4 = 1
c = 96146522937.2138934755566928890704687838407524... if n mod 4 = 2
c = 96146521894.8218716328341714149619262713426755... if n mod 4 = 3
(End)

a(0)=1 prepended by Alois P. Heinz, Nov 26 2015

A279785 Number of ways to choose a strict partition of each part of a strict partition of n.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 11, 18, 28, 47, 71, 108, 166, 252, 382, 587, 869, 1282, 1938, 2832, 4153, 6148, 8962, 12965, 18913, 27301, 39380, 56747, 81226, 115907, 166358, 236000, 334647, 475517, 671806, 947552, 1335679, 1875175, 2630584, 3687589, 5150585, 7183548

Offset: 0

Gus Wiseman, Dec 18 2016

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000009(n). - Seiichi Manyama, Nov 14 2018

			The a(6)=11 twice-partitions are:
((6)), ((5)(1)), ((51)), ((4)(2)), ((42)), ((41)(1)),
((3)(2)(1)), ((31)(2)), ((32)(1)), ((321)), ((21)(2)(1)).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
Gus Wiseman, Sequences enumerating triangles of integer partitions

Cf. A000009, A063834, A270995, A279375, A327553, A327605.

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, g(i)*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016

nn=20;CoefficientList[Series[Product[(1+PartitionsQ[k]x^k),{k,nn}],{x,0,nn}],x]
(* Second program: *)
g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n]; b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, g[i]*b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

G.f.: Product_{k>0} (1 + A000009(k)*x^k). - Seiichi Manyama, Nov 14 2018

A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 3, 0, 4, 1, 1, 0, 5, 0, 6, 0, 2, 2, 8, 0, 2, 2, 0, 0, 10, 1, 12, 0, 2, 3, 2, 0, 15, 3, 2, 0, 18, 1, 22, 0, 0, 5, 27, 0, 2, 0, 3, 0, 32, 0, 3, 0, 4, 5, 38, 0, 46, 7, 0, 0, 4, 1, 54, 0, 5, 1, 64, 0, 76, 8, 0, 0, 3, 1, 89, 0, 0, 10

Offset: 1

Gus Wiseman, May 15 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 25 are (3,3), for which we have choices ((3),(2,1)) and ((2,1),(3)), so a(25) = 2.
The prime indices of 91 are (4,6), for which we have choices ((4),(6)), ((4),(5,1)), ((4),(3,2,1)), ((3,1),(6)), ((3,1),(4,2)), so a(91) = 5.
The prime indices of 273 are (2,4,6), for which we have choices ((2),(4),(6)), ((2),(4),(5,1)), ((2),(3,1),(6)), so a(273) = 3.

Adding up over all integer partitions gives A279790, strict A279375.
Without disjointness we have A357982, non-strict version A299200.
For multiplicities instead of indices we have A382525.
Positions of 0 appear to be A382912, counted by A383710, odd case A383711.
Positions of positive terms are A382913, counted by A383708, odd case A383533.
Positions of 1 are A383707, counted by A179009.
The conjugate version is A384005.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A098859, A122111, A130091, A209229, A317141, A381454, A382771, A382876.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[pof[prix[n]]],{n,100}]

A296122 Number of twice-partitions of n with no repeated partitions.

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 40, 77, 157, 285, 552, 1018, 1921, 3484, 6436, 11622, 21082, 37550, 67681, 119318, 211792, 372003, 653496, 1137185, 1986234, 3429650, 5935970, 10205907, 17537684, 29958671, 51189932, 86967755, 147759421, 249850696, 422123392, 710495901

Offset: 0

Gus Wiseman, Dec 05 2017

nonn

a(n) is the number of sequences of distinct integer partitions whose sums are weakly decreasing and add up to n.

			The a(4) = 10 twice-partitions: (4), (31), (22), (211), (1111), (3)(1), (21)(1), (111)(1), (2)(11), (11)(2).

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

Cf. A000009, A000041, A047968, A063834, A261049, A273873, A279375, A295279, A296121.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 06 2017

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@p],UnsameQ@@#&],{p,IntegerPartitions[n]}]],{n,15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[j!*
     Binomial[PartitionsP[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(15)-a(34) from Robert G. Wilson v, Dec 06 2017

A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 28, 51, 92, 164, 289, 504, 871, 1493, 2539, 4290, 7201, 12017, 19939, 32911, 54044, 88330, 143709, 232817, 375640, 603755, 966816, 1542776, 2453536, 3889338, 6146126, 9683279, 15211881, 23830271, 37230720, 58015116, 90174847, 139820368, 216286593

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

Also the number of integer compositions of n whose leaders of maximal weakly decreasing runs are strictly increasing. For example, the composition (1,2,2,1,3,1,4,1) has maximal weakly decreasing runs ((1),(2,2,1),(3,1),(4,1)), with leaders (1,2,3,4), so is counted under a(15). - Gus Wiseman, Aug 21 2024

			The a(1) = 1 through a(5) = 15 multiset partitions:
  {1}  {2}    {3}        {4}          {5}
       {1,1}  {1,2}      {1,3}        {1,4}
              {1,1,1}    {2,2}        {2,3}
              {1},{1,1}  {1,1,2}      {1,1,3}
                         {1,1,1,1}    {1,2,2}
                         {1},{1,2}    {1,1,1,2}
                         {2},{1,1}    {1},{1,3}
                         {1},{1,1,1}  {1},{2,2}
                                      {2},{1,2}
                                      {3},{1,1}
                                      {1,1,1,1,1}
                                      {1},{1,1,2}
                                      {2},{1,1,1}
                                      {1},{1,1,1,1}
                                      {1,1},{1,1,1}
From _Gus Wiseman_, Aug 21 2024: (Start)
The a(0) = 1 through a(5) = 15 compositions whose leaders of maximal weakly decreasing runs are strictly increasing:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The version for set partitions is A007837.
For sums instead of sizes we have A271619.
For constant instead of distinct sizes we have A319066.
These multiset partitions are ranked by A326533.
For odd instead of distinct sizes we have A356932.
The version for twice-partitions is A358830.
The case of distinct sums also is A358832.
Ranked by positions of strictly increasing rows in A374740, opposite A374629.
A001970 counts multiset partitions of integer partitions.
A011782 counts compositions.
A063834 counts twice-partitions, strict A296122.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Runs and leaders: A000041, A188900, A374634, A374635, A374637, A374679, A374742 (ranks A374744), A374743 (ranks A374701), A374746, A374747.
Cf. A000009, A141199, A273873, A279375, A279785, A279790, A358334.
Cf. A003242, A106356, A188920, A189076, A238343, A333213, A336342.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@Length/@#&]],{n,0,10}]
(* second program *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 21 2024 *)

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, 1 + polcoef(g, k, y) + O(x*x^n)))} \\ Andrew Howroyd, Dec 31 2022

G.f.: Product_{k>=1} (1 + [y^k]P(x,y)) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022

A279789 Number of ways to choose a constant partition of each part of a constant partition of n.

Original entry on oeis.org

1, 1, 3, 3, 8, 3, 17, 3, 30, 12, 41, 3, 130, 3, 137, 45, 359, 3, 656, 3, 1306, 141, 2057, 3, 5446, 36, 8201, 544, 18610, 3, 34969, 3, 72385, 2061, 131081, 165, 290362, 3, 524297, 8205, 1109206, 3, 2130073, 3, 4371490, 33594, 8388617, 3, 17445321, 132, 33556496

Offset: 0

Gus Wiseman, Dec 18 2016

nonn

Also number of ways to choose a divisor d|n and then a sequence of n/d divisors of d.

			The a(6)=17 twice-constant partitions are:
((6)),
((3)(3)), ((33)),
((3)(111)), ((111)(3)),
((2)(2)(2)), ((222)),
((2)(2)(11)), ((2)(11)(2)), ((11)(2)(2)),
((2)(11)(11)), ((11)(2)(11)), ((11)(11)(2)),
((1)(1)(1)(1)(1)(1)), ((11)(11)(11)), ((111)(111)), ((111111)).

Alois P. Heinz, Table of n, a(n) for n = 0..6643
Gus Wiseman, Sequences enumerating triangles of integer partitions

Cf: A000005 (tau), A000040, A018818, A063834, A260685, A279375.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(tau(n/d)^d, d=divisors(n)))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016

nn=20;Table[DivisorSum[n,Power[DivisorSigma[0,#],n/#]&],{n,nn}]

a(n)=if(n==0, 1, sumdiv(n, d, numdiv(n/d)^d)) \\ Andrew Howroyd, Aug 26 2018

a(n) = Sum_{d|n} tau(n/d)^d for n > 0. - Andrew Howroyd, Aug 26 2018
G.f.: 1 + Sum_{k>=1} tau(k)*x^k/(1 - tau(k)*x^k). - Ilya Gutkovskiy, May 23 2019
a(n) = 3 <=> n is prime <=> n in { A000040 }. - Alois P. Heinz, May 23 2019

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

Original entry on oeis.org

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

A384321 Numbers whose distinct prime indices are not maximally refined.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 111, 113, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jun 01 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Given a partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

			The prime indices of 25 are {3,3}, which has refinements: ((3),(1,2)) and ((1,2),(3)), so 25 is in the sequence.
The prime indices of 102 are {1,2,7}, which has refinement ((1),(2),(3,4)), so 102 is in the sequence.
The terms together with their prime indices begin:
     5: {3}      39: {2,6}      73: {21}
     7: {4}      41: {13}       74: {1,12}
    11: {5}      43: {14}       77: {4,5}
    13: {6}      46: {1,9}      79: {22}
    17: {7}      47: {15}       82: {1,13}
    19: {8}      49: {4,4}      83: {23}
    21: {2,4}    51: {2,7}      85: {3,7}
    22: {1,5}    53: {16}       86: {1,14}
    23: {9}      55: {3,5}      87: {2,10}
    25: {3,3}    57: {2,8}      89: {24}
    26: {1,6}    58: {1,10}     91: {4,6}
    29: {10}     59: {17}       93: {2,11}
    31: {11}     61: {18}       94: {1,15}
    33: {2,5}    62: {1,11}     95: {3,8}
    34: {1,7}    65: {3,6}      97: {25}
    35: {3,4}    67: {19}      101: {26}
    37: {12}     69: {2,9}     102: {1,2,7}
    38: {1,8}    71: {20}      103: {27}

These appear to be positions of terms > 1 in A383706, non-disjoint A357982, non-strict A299200.
The strict complement is A383707, counted by A179009.
Partitions of this type appear to be counted by A384317.
The complement is A384320.
The strict (squarefree) case appears to be A384322, counted by A384318.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
Cf. A098859, A122111, A130091, A317142, A326080, A381454, A382525, A384005, A384323, A384390.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[30],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]!={}]&]

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 37, 57, 89, 265, 309, 521, 745, 1129, 3005, 3545, 5685, 8201, 12265, 16629, 41369, 48109, 77265, 107645, 160681, 214861, 316913, 644837, 798861, 1207445, 1694269, 2437689, 3326705, 4710397, 6270513, 12246521, 14853625, 22244569, 30308033, 43706705, 57926577, 82166105, 107873221, 148081785, 257989961, 320873065, 458994657, 628016225, 875485585, 1165065733

Offset: 0

Gus Wiseman, Jul 10 2020

nonn

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

			The a(0) = 1 through a(5) = 5 splits:
  ()  (1)  (2)  (3)     (4)     (5)
                (12)    (13)    (14)
                (21)    (31)    (23)
                (1)(2)  (1)(3)  (32)
                (2)(1)  (3)(1)  (41)
                                (1)(4)
                                (2)(3)
                                (3)(2)
                                (4)(1)
The a(6) = 29 splits:
  (6)    (1)(5)   (1)(2)(3)
  (15)   (2)(4)   (1)(3)(2)
  (24)   (4)(2)   (2)(1)(3)
  (42)   (5)(1)   (2)(3)(1)
  (51)   (1)(23)  (3)(1)(2)
  (123)  (1)(32)  (3)(2)(1)
  (132)  (13)(2)
  (213)  (2)(13)
  (231)  (2)(31)
  (312)  (23)(1)
  (321)  (31)(2)
         (32)(1)

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

The version with equal instead of different sums is A336130.
Starting with a non-strict composition gives A336127.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Set partitions with distinct block-sums are A275780.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, 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],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(50) from Max Alekseyev, Feb 14 2024

A300301 Number of ways to choose a partition, with odd parts, of each part of a partition of n into odd parts.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 15, 21, 37, 56, 80, 127, 183, 280, 428, 616, 893, 1367, 1944, 2846, 4223, 6049, 8691, 12670, 18128, 25921, 37529, 53338, 75738, 108561, 153460, 216762, 308829, 433893, 612006, 864990, 1211097, 1697020, 2386016, 3331037, 4648229, 6503314

Offset: 0

Gus Wiseman, Mar 02 2018

nonn

			The a(6) = 10 twice-partitions using odd partitions: (5)(1), (3)(3), (113)(1), (3)(111), (111)(3), (3)(1)(1)(1), (11111)(1), (111)(111), (111)(1)(1)(1), (1)(1)(1)(1)(1)(1).

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

A300300(n) <= a(n) <= A279785(n)

Cf. A000009, A063834, A078408, A089259, A270995, A271619, A279374, A279375, A279790, A294617, A300300.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-2)+`if`(i>n, 0, b(i)*g(n-i, i)))
    end:
a:= n-> g(n, n-1+irem(n,2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 05 2018

nn=50;
ser=Product[1/(1-PartitionsQ[n]x^n),{n,1,nn,2}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

O.g.f.: Product_{n odd} 1/(1 - A000009(n)x^n).

cp's OEIS Frontend

A063834 Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.

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A279785 Number of ways to choose a strict partition of each part of a strict partition of n.

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A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

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A296122 Number of twice-partitions of n with no repeated partitions.

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A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

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A279789 Number of ways to choose a constant partition of each part of a constant partition of n.

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

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