A063834 -id:A063834 - cp's OEIS frontend

A358901 Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 7, 9, 8, 9, 11, 11, 15, 16, 16, 18, 20, 22, 26, 28, 31, 32, 36, 40, 45, 46, 46, 50, 59, 64, 70, 75, 78, 83, 89, 94, 108, 106, 104, 120, 137, 142, 147, 150, 161, 174, 190, 200, 220, 226, 224, 248, 274, 274, 287, 301, 320, 340, 351, 361

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(11) = 7 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (41)  (42)  (43)   (62)   (54)   (82)   (74)
                              (51)  (61)   (71)   (63)   (91)   (65)
                                    (421)  (431)  (81)   (451)  (83)
                                                  (621)  (631)  (92)
                                                                (A1)
                                                                (821)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

The weakly decreasing version is A358909 (complement A358910).
The version not counting multiplicity is A358903, weakly decreasing A358902.
For equal numbers of prime factors we have A319169, compositions A358911.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A056239, A129519, A141199, A218482, A300335, A319071, A320324, A358335, A358831, A358908.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@PrimeOmega/@#&]],{n,0,60}]

a(61) and beyond from Lucas A. Brown, Dec 14 2022

A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 49, 91, 179, 341, 664, 1280, 2503, 4872, 9557, 18750, 36927, 72800, 143880, 284660, 564093, 1118911, 2221834, 4415417, 8781591, 17476099, 34799199, 69327512, 138176461, 275503854, 549502119, 1096327380, 2187894634, 4367310138, 8719509111

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(0) = 1 through a(4) = 13 sequences:
  ()  ((1))  ((2))     ((3))        ((4))
             ((11))    ((21))       ((22))
             ((1)(1))  ((111))      ((31))
                       ((1)(2))     ((211))
                       ((2)(1))     ((1111))
                       ((1)(1)(1))  ((1)(3))
                                    ((2)(2))
                                    ((3)(1))
                                    ((11)(11))
                                    ((1)(1)(2))
                                    ((1)(2)(1))
                                    ((2)(1)(1))
                                    ((1)(1)(1)(1))

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

The case of set partitions is A038041.
The version for weakly decreasing lengths is A141199, strictly A358836.
For equal sums instead of lengths we have A279787.
The case of twice-partitions is A306319, distinct A358830.
The unordered version is A319066.
The case of plane partitions is A323429.
The case of constant sums also is A358833.
A055887 counts sequences of partitions with total sum n.
A281145 counts same-trees.
A319169 counts partitions with constant Omega, ranked by A320324.
A358911 counts compositions with constant Omega, distinct A358912.
Cf. A000041, A000219, A001970, A063834, A218482, A305551, A358835.

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],SameQ@@Length/@#&]],{n,0,10}]

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

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

Terms a(16) and beyond from Andrew Howroyd, Dec 31 2022

A358906 Number of finite sequences of distinct integer partitions with total sum n.

Original entry on oeis.org

1, 1, 2, 7, 13, 35, 87, 191, 470, 1080, 2532, 5778, 13569, 30715, 69583, 160386, 360709, 814597, 1824055, 4102430, 9158405, 20378692, 45215496, 100055269, 221388993, 486872610, 1069846372, 2343798452, 5127889666, 11186214519, 24351106180, 52896439646

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(4) = 13 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((211))
                 ((2)(1))   ((1111))
                 ((1)(11))  ((1)(3))
                 ((11)(1))  ((3)(1))
                            ((11)(2))
                            ((1)(21))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))

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

This is the case of A055887 with distinct partitions.
The unordered version is A261049.
The case of twice-partitions is A296122.
The case of distinct sums is A336342, constant sums A279787.
The version for sequences of compositions is A358907.
The case of weakly decreasing lengths is A358908.
The case of distinct lengths is A358912.
The version for strict partitions is A358913, distinct case of A304969.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all distinct Omegas.
Cf. A000009, A000041, A000219, A098407, A271619, A330463, A358836, A358901, A358914.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1, p+j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 13 2024

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

a(n) = Sum_{k} A330463(n,k) * k!.

A300352 Number of strict trees of weight n with distinct leaves.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 11, 17, 40, 48, 76, 109, 159, 400, 470, 745, 1057, 1576, 2103, 5267, 6022, 9746, 13390, 20099, 26542, 39396, 82074, 101387, 152291, 215676, 308937, 423587, 596511, 799022, 1623311, 1960223, 2947722, 4048704, 5845982, 7794809, 11028888

Offset: 1

Gus Wiseman, Mar 03 2018

nonn

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) = 11 strict trees with distinct leaves: 8, (71), ((52)1), ((43)1), (62), ((51)2), (53), ((41)3), (5(21)), (521), (431).

Cf. A000009, A056239, A063834, A196545, A246867, A273873, A281145, A289501, A294018, A294079, A296150, A299201, A299202, A299203, A300353, A300354, A300355.

sps[{}]:={{}};sps[set:{i_,_}]:=
Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
str[q_]:=str[q]=If[Length[q]===1,1,Total[Times@@@Map[str,Select[sps[q],And[Length[#]>1,UnsameQ@@Total/@#]&],{2}]]];
Table[Total[str/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,20}]

a(n) = Sum_{i=1..A000009(n)} A294018(A246867(n,i)).

A300442 Number of binary strict trees of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 23, 46, 108, 231, 561, 1285, 3139, 7348, 18265, 43907, 109887, 267582, 675866, 1669909, 4238462, 10555192, 26955062, 67706032, 173591181, 438555624, 1129088048, 2869732770, 7410059898, 18911818801, 48986728672, 125562853003, 326011708368

Offset: 0

Gus Wiseman, Mar 05 2018

nonn

A binary strict tree of weight n > 0 is either a single node of weight n, or an ordered pair of binary strict trees with strictly decreasing weights summing to n.

			The a(5) = 6 binary strict trees: 5, (41), (32), ((31)1), ((21)2), (((21)1)1).
The a(6) = 10 binary strict trees:
  6,
  (51), (42),
  ((41)1), ((32)1), ((31)2),
  (((31)1)1), (((21)2)1), (((21)1)2),
  ((((21)1)1)1).

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

Cf. A000992, A001190, A063834, A196545, A273873, A289501, A292432, A293511, A300352, A300440, A300443.

a:= proc(n) option remember;
      1+add(a(j)*a(n-j), j=1..(n-1)/2)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 06 2018

k[n_]:=k[n]=1+Sum[Times@@k/@y,{y,Select[IntegerPartitions[n],Length[#]===2&&UnsameQ@@#&]}];
Array[k,40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n - j], {j, 1, (n - 1)/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

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

a(n) = 1 + Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).

A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 4, 5, 3, 1, 3, 7, 12, 12, 6, 1, 3, 9, 19, 28, 25, 11, 1, 4, 14, 36, 65, 81, 63, 24, 1, 4, 16, 48, 107, 172, 193, 136, 47, 1, 5, 22, 75, 192, 369, 522, 522, 331, 103, 1, 5, 25, 96, 284, 643, 1108, 1420, 1292, 750, 214, 1, 6

Offset: 1

Gus Wiseman, Mar 19 2018

A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.

			Triangle begins:
  1
  1   1
  1   1   1
  1   2   3   2
  1   2   4   5   3
  1   3   7  12  12   6
  1   3   9  19  28  25  11
  1   4  14  36  65  81  63  24
  1   4  16  48 107 172 193 136  47
  1   5  22  75 192 369 522 522 331 103
  ...
The T(6,3) = 7 binary enriched p-trees: ((41)1), ((32)1), (4(11)), ((31)2), ((22)2), (3(21)), ((21)3).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Last entries of each row give A000992. Row sums are A300443.

Cf. A001190, A008284, A055277, A063834, A196545, A273873, A289501, A292050, A298422, A298426, A300354, A300439, A300442, A301344, A301364-A301367.

bintrees[n_]:=Prepend[Join@@Table[Tuples[bintrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]===2&]}],n];
Table[Length[Select[bintrees[n],Count[#,_Integer,{-1}]===k&]],{n,13},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sum(k=1, n\2, v[k]*v[n-k])); apply(p->Vecrev(p/y), v)}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

A302493 Prime numbers of prime-power index.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 31, 41, 53, 59, 67, 83, 97, 103, 109, 127, 131, 157, 179, 191, 211, 227, 241, 277, 283, 311, 331, 353, 367, 401, 419, 431, 461, 509, 547, 563, 587, 599, 617, 661, 691, 709, 719, 739, 773, 797, 859, 877, 919, 967, 991, 1009, 1031

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

Cf. A000040, A000961, A001222, A005117, A007097, A056239, A063834, A275024, A281113, A302242, A302491, A302492.

Prime/@Select[Range[100],Or[#===1,PrimePowerQ[#]]&]

forprime(p=1, 500, if(p==2 || isprimepower(primepi(p)), print1(p, ", "))) \\ Felix Fröhlich, Apr 10 2018

a(n) = A000040(A000961(n)).

A318434 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(3072) = 5 constant-sum split partitions:
  (21111111111)
  (21111)(111111)
  (211)(1111)(1111)
  (21)(111)(111)(111)
  (2)(11)(11)(11)(11)(11)

Cf. A001970, A056239, A063834, A296150, A316223, A317545, A317546, A319002.

Cf. A316245, A317508, A317715, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Length[Select[comps[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],SameQ@@Total/@#&]],{n,100}]

A318948 Number of ways to choose an integer partition of each factor in a factorization of n.

Original entry on oeis.org

1, 2, 3, 9, 7, 17, 15, 40, 39, 56, 56, 126, 101, 165, 197, 336, 297, 496, 490, 774, 837, 1114, 1255, 1948, 2007, 2638, 3127, 4123, 4565, 6201, 6842, 9131, 10311, 12904, 14988, 19516, 21637, 26995, 31488, 39250, 44583, 55418, 63261, 77683, 89935, 108068, 124754

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

			The a(4) = 9 ways: (1+1)*(1+1), (1+1+1+1), (1+1)*(2), (2)*(1+1), (2+1+1), (2)*(2), (2+2), (3+1), (4).

Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A121229, A281113, A284639, A318949.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@PartitionsP/@fac,{fac,facs[n]}],{n,10}]

Dirichlet g.f.: Product_{n > 1} 1 / (1 - P(n) / n^s) where P = A000041. [clarified by Ilya Gutkovskiy, Oct 26 2019]

A330462 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 2, 0, 0, 0, 0, 5, 11, 3, 0, 0, 0, 0, 0, 6, 16, 8, 0, 0, 0, 0, 0, 0, 8, 25, 15, 1, 0, 0, 0, 0, 0, 0, 10, 35, 28, 4, 0, 0, 0, 0, 0, 0, 0, 12, 52, 46, 9, 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  2  0  0
  0  3  4  0  0  0
  0  4  6  2  0  0  0
  0  5 11  3  0  0  0  0
  0  6 16  8  0  0  0  0  0
  0  8 25 15  1  0  0  0  0  0
  0 10 35 28  4  0  0  0  0  0  0
  ...
Row n = 7 counts the following set-systems:
  {{7}}      {{1},{6}}      {{1},{2},{4}}
  {{1,6}}    {{2},{5}}      {{1},{2},{1,3}}
  {{2,5}}    {{3},{4}}      {{1},{3},{1,2}}
  {{3,4}}    {{1},{1,5}}
  {{1,2,4}}  {{1},{2,4}}
             {{2},{1,4}}
             {{2},{2,3}}
             {{3},{1,3}}
             {{4},{1,2}}
             {{1},{1,2,3}}
             {{1,2},{1,3}}

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

Row sums are A050342.
Column k = 1 is A000009.
Cf. A001970, A050343, A063834, A270995, A271619, A279375, A279785, A283877, A294617, A326031, A330456, A330460, A330463, A360764.

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@@#,And@@UnsameQ@@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]

L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
A(n)={my(c=L(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c,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)^A000009(j). - Andrew Howroyd, Dec 29 2019

cp's OEIS Frontend

A358901 Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).

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A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

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A358906 Number of finite sequences of distinct integer partitions with total sum n.

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A300352 Number of strict trees of weight n with distinct leaves.

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A300442 Number of binary strict trees of weight n.

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A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

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Mathematica

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A302493 Prime numbers of prime-power index.

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A318434 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with equal sums.

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