A279785 -id:A279785 - cp's OEIS frontend

A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

1, 1, 2, 4, 9, 19, 37, 76, 159, 326, 671, 1376, 2815, 5759, 11774, 24083, 49249, 100632, 205490, 419420, 855799, 1745889, 3561867, 7268240, 14836127, 30295633, 61888616

Offset: 0

Gus Wiseman, Mar 26 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(4) = 9 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1},{2,2,2}}
                                   {{2},{1,1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{1},{2},{3},{4}}
The a(5) = 19 factorizations:
  32  2*16  2*3*27   2*3*5*25  2*3*5*7*11
      4*8   2*4*9    2*3*5*9
      2*81  2*3*8    2*3*5*49
      4*27  2*3*125  2*3*7*25
      9*8   2*9*25
      3*16  2*5*27
            5*4*9

Without distinct sums we have A055887.
Twice-partitions of this type are counted by A279786.
For distinct blocks instead of sums we have A304969.
Without constant blocks we have A326519.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A381718.
For equal instead of distinct block-sums we have A382204.
For equal block-sums and strict blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A116540, A255906, A317532.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Cf. A007716, A116539, A255903, A275780, A317583, A326517, A326518, A381633, A381636, A382216, A382428.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(14)-a(26) from Christian Sievers, Apr 04 2025

A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.

Original entry on oeis.org

1, 1, 1, 6, 8, 35, 292, 673, 2818, 16956, 219772, 636748, 3768505, 20309534, 183403268, 3227600747, 12272598308, 81353466578, 561187259734, 4416808925866, 50303004612136, 1238783066956740, 5566249468690291, 44970939483601100, 330144217684933896, 3131452652308459402

Offset: 0

Gus Wiseman, Mar 29 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(4) = 8 multiset partitions:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}
                  {{1},{1,2}}  {{1},{1,2,3}}
                  {{1},{2,3}}  {{1},{2,3,4}}
                  {{2},{1,2}}  {{2},{1,2,3}}
                  {{2},{1,3}}  {{2},{1,3,4}}
                  {{3},{1,2}}  {{3},{1,2,3}}
                               {{3},{1,2,4}}
                               {{4},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

For distinct sums instead of sizes we have A116539, see A050326.
Without distinct lengths we have A116540 (normal set multipartitions).
Without strict blocks we have A326517, for sum instead of size A326519.
For equal instead of distinct sizes we have A331638.
Twice-partitions of this type are counted by A358830.
For distinct sums instead of sizes we have A381718.
For equal instead of distinct sizes we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A255903, A275780, A279785, A317532, A326518, A381633, A382214, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Length/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Mar 31 2025

a(10) onwards from Andrew Howroyd, Mar 31 2025

A327605 Number of parts in all twice partitions of n where both partitions are strict.

Original entry on oeis.org

0, 1, 1, 5, 8, 15, 28, 49, 86, 156, 259, 412, 679, 1086, 1753, 2826, 4400, 6751, 10703, 16250, 24757, 38047, 57459, 85861, 129329, 192660, 286177, 424358, 624510, 915105, 1347787, 1961152, 2847145, 4144089, 5988205, 8638077, 12439833, 17837767, 25536016

Offset: 0

Alois P. Heinz, Sep 18 2019

nonn

			a(3) = 5 = 1+2+2 counting the parts in 3, 21, 2|1.

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

Cf. A000009, A279785, A327553, A327594, A327607, A327608, A327795.

Column k=2 of A327622.

g:= proc(n, i) option remember; `if`(i*(i+1)/2 f+
       [0, f[1]])(g(n-i, min(n-i, i-1)))))
    end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 (f-> f+[0, f[1]*
       h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i$2))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..42);

b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, Return@{1, 0}]; If[k == 0, Return@{1, 1}]; If[i (i + 1)/2 < n, Return@{0, 0}]; b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][b[i, i, k - 1]]];
a[n_] := b[n, n, 2][[2]];
a /@ Range[0, 42] (* Jean-François Alcover, Jun 03 2020, after Alois P. Heinz in A327622 *)

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).

A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 0, 7, 11, 1, 0, 0, 0, 11, 20, 6, 0, 0, 0, 0, 15, 40, 16, 0, 0, 0, 0, 0, 22, 68, 40, 3, 0, 0, 0, 0, 0, 30, 120, 91, 11, 0, 0, 0, 0, 0, 0, 42, 195, 186, 41, 0, 0, 0, 0, 0, 0, 0, 56, 320, 367, 105, 3, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
  1
  0  1
  0  2  0
  0  3  2  0
  0  5  4  0  0
  0  7 11  1  0  0
  0 11 20  6  0  0  0
  0 15 40 16  0  0  0  0
  0 22 68 40  3  0  0  0  0
  ...
Row n = 5 counts the following sets of multisets:
  {{5}}          {{1},{4}}        {{1},{2},{1,1}}
  {{1,4}}        {{2},{3}}
  {{2,3}}        {{1},{1,3}}
  {{1,1,3}}      {{1},{2,2}}
  {{1,2,2}}      {{2},{1,2}}
  {{1,1,1,2}}    {{3},{1,1}}
  {{1,1,1,1,1}}  {{1},{1,1,2}}
                 {{1,1},{1,2}}
                 {{2},{1,1,1}}
                 {{1},{1,1,1,1}}
                 {{1,1},{1,1,1}}

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

Row sums are A261049.
Column k = 1 is A000041.
Multisets of multisets are A061260, with row sums A001970.
Sets of sets are A330462, with row sums A050342.
Multisets of sets are A285229, with row sums A089259.
Sets of disjoint sets are A330460, with row sums A294617.
Cf. A007716, A060016, A063834, A255906, A271619, A279785, A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
       combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^j), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 30 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],And[UnsameQ@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
     PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} (1 + y*x^j)^A000041(j). - Andrew Howroyd, Dec 29 2019

A300353 Number of strict trees of weight n with odd leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 2, 4, 7, 14, 24, 46, 92, 186, 368, 750, 1529, 3160, 6510, 13590, 28374, 59780, 125732, 266468, 564188, 1202842, 2560106, 5484304, 11732400, 25229068, 54187918, 116938702, 252039411, 545593378, 1179545874, 2560009400, 5550315640, 12075064432

Offset: 0

Gus Wiseman, Mar 03 2018

nonn

This sequence is initially dominated by A300352 but eventually becomes much greater.

A strict tree of weight n > 0 is either a single node of weight n, or a sequence of two or more strict trees with strictly decreasing weights summing to n.

			The a(8) = 7 strict trees with odd leaves: (71), (53), (((51)1)1), (((31)3)1), (((31)1)3), ((31)31), (((((31)1)1)1)1).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294018, A294079, A299203, A300300, A300301, A300351, A300352, A300354, A300355.

d[n_]:=d[n]=If[EvenQ[n],0,1]+Sum[Times@@d/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}];
Table[d[n],{n,40}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(x/(1-x^2) + prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018

O.g.f: (1 + x/(1-x^2) + Product_{i>0} (1 + a(i)x^i))/2.

a(n) = Sum_{i=1..A000009(n)} A294018(A300351(n,i)).

A300355 Number of enriched p-trees of weight n with odd leaves.

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 47, 132, 410, 1254, 4052, 12818, 42783, 139082, 469924, 1563606, 5353966, 18065348, 62491018, 213391790, 743836996, 2565135934, 8994087070, 31251762932, 110245063771, 385443583008, 1365151504722, 4800376128986, 17070221456536, 60289267885410

Offset: 0

Gus Wiseman, Mar 03 2018

nonn

An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

			The a(5) = 16 enriched p-trees of weight with odd leaves:
5,
((31)1), ((((11)1)1)1), (((111)1)1), (((11)(11))1), (((11)11)1), ((1111)1),
(3(11)), (((11)1)(11)), ((111)(11)),
(311), (((11)1)11), ((111)11),
((11)(11)1),
((11)111),
(11111).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294079, A299202, A299203, A300300, A300301, A300352, A300353, A300354.

c[n_]:=c[n]=If[EvenQ[n],0,1]+Sum[Times@@c/@y,{y,Select[IntegerPartitions[n],Length[#]>1&]}];
Table[c[n],{n,30}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = n%2 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

O.g.f: (1 + x/(1-x^2) + Prod_{i>0} 1/(1 - a(i)x^i))/2.

a(n) = Sum_{i=1..A000009(n)} A299203(A300351(n,i)).

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

Original entry on oeis.org

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

A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 10, 13, 15, 22, 20, 32, 32, 43, 49, 65, 64, 92, 96, 121, 140, 173, 192

Offset: 0

Gus Wiseman, Mar 29 2025

			The partition y = (3,3,2,1,1,1) has 2 partitions into sets: {{1},{3},{1,2},{1,3}} and {{1},{1,3},{1,2,3}}, but only the latter has distinct sums, so y is counted under a(11)
The a(1) = 1 through a(10) = 10 partitions (A=10):
  1  2  3  4    5    6     7    8      9      A
           211  221  411   322  332    441    433
                311  2211  331  422    522    442
                           511  611    711    622
                                3311   42111  811
                                32111         3322
                                              4411
                                              32221
                                              43111
                                              52111

Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633.
Normal multiset partitions of this type are counted by A381718.
These partitions are ranked by A381870.
For no choices we have A381990, ranks A381806, see A382078, ranks A293243.
For at least one choice we have A381992, ranks A382075, see A382077, ranks A382200.
For distinct blocks instead of block-sums we have A382079, ranks A293511.
MM-numbers of these multiset partitions are A382201, see A302478.
For constant instead of strict blocks we have A382301, ranks A381991.
Set systems: A050320, A050326, A050342, A116539, A296120, A318361.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A299202, A317142, A358914, A381441, A381454, A381636.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&) /@ Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n], Length[Select[ssfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,15}]

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).

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