A271619 -id:A271619 - cp's OEIS frontend

A358826 Number of ways to choose a sequence of partitions, one of each part of an odd-length partition of 2n+1 into odd parts.

1, 4, 11, 35, 113, 326, 985, 3124, 8523, 24519, 71096, 191940, 530167, 1442059, 3833007, 10243259, 27151086, 71032191, 184492464, 478339983, 1227208513, 3140958369, 8016016201, 20210235189, 50962894061, 127936646350, 319022819270, 794501931062, 1969154638217

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

			The a(1) = 1 through a(5) = 11 twice-partitions:
  (1)  (3)        (5)
       (21)       (32)
       (111)      (41)
       (1)(1)(1)  (221)
                  (311)
                  (2111)
                  (11111)
                  (3)(1)(1)
                  (21)(1)(1)
                  (111)(1)(1)
                  (1)(1)(1)(1)(1)

Gus Wiseman, Sequences enumerating triangles of integer partitions

For odd parts instead of length and sums we have A270995.
Requiring odd lengths and odd parts gives A279374 aerated.
This is the case of A358824 with odd sums.
This is the odd-length case (hence odd bisection) of A358825.
For odd lengths (instead of length) we have A358827.
For odd lengths instead of sums we have A358834.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A072233, A271619, A279785, A356932.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Total/@#]&]],{n,1,15,2}]

A358827 Number of twice-partitions of n into partitions with all odd lengths and sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 11, 19, 27, 51, 83, 128, 208, 324, 542, 856, 1332, 2047, 3371, 5083, 8009, 12545, 19478, 29770, 46038, 70777, 108627, 167847, 255408, 388751, 593475, 901108, 1361840, 2077973, 3125004, 4729056, 7146843, 10732799, 16104511, 24257261, 36305878

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(6) = 11 twice-partitions:
  (1)  (1)(1)  (3)        (3)(1)        (5)              (3)(3)
               (111)      (111)(1)      (221)            (5)(1)
               (1)(1)(1)  (1)(1)(1)(1)  (311)            (111)(3)
                                        (11111)          (221)(1)
                                        (3)(1)(1)        (3)(111)
                                        (111)(1)(1)      (311)(1)
                                        (1)(1)(1)(1)(1)  (111)(111)
                                                         (11111)(1)
                                                         (3)(1)(1)(1)
                                                         (111)(1)(1)(1)
                                                         (1)(1)(1)(1)(1)(1)

Gus Wiseman, Sequences enumerating triangles of integer partitions

This is the case of A358334 with odd sums.
This is the case of A358825 with odd lengths.
The case of odd length is the odd bisection.
For odd parts instead of lengths and sums we have A270995.
Requiring odd parts also gives A279374 aerated.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A072233, A271619, A279785, A306319, A356932, A358824.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&&OddQ[Times@@Total/@#]&]],{n,0,10}]

G.f.: Product_{k odd} 1/(1-A027193(k)*x^k).

A358913 Number of finite sequences of distinct sets with total sum n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 28, 45, 86, 172, 344, 608, 1135, 2206, 4006, 7689, 13748, 25502, 47406, 86838, 157560, 286642, 522089, 941356, 1718622, 3079218, 5525805, 9902996, 17788396, 31742616, 56694704, 100720516, 178468026, 317019140, 560079704, 991061957

Offset: 0

Gus Wiseman, Dec 11 2022

nonn

			The a(1) = 1 through a(5) = 11 sequences of sets:
  ({1})  ({2})  ({3})      ({4})        ({5})
                ({1,2})    ({1,3})      ({1,4})
                ({1},{2})  ({1},{3})    ({2,3})
                ({2},{1})  ({3},{1})    ({1},{4})
                           ({1},{1,2})  ({2},{3})
                           ({1,2},{1})  ({3},{2})
                                        ({4},{1})
                                        ({1},{1,3})
                                        ({1,2},{2})
                                        ({1,3},{1})
                                        ({2},{1,2})

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

The unordered version is A050342, non-strict A261049.
The case of strictly decreasing sums is A279785.
This is the distinct case of A304969.
The case of distinct sums is A336343, constant sums A279791.
This is the case of A358906 with strict partitions.
The version for compositions instead of strict partitions is A358907.
The case of twice-partitions is A358914.
A001970 counts multiset partitions of integer partitions.
A055887 counts sequences of partitions.
A063834 counts twice-partitions.
A330462 counts set systems by total sum and length.
A358830 counts twice-partitions with distinct lengths.
Cf. A000009, A000041, A000219, A271619, A296122, A336342, A358908.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(binomial(g(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..35);  # Alois P. Heinz, Feb 13 2024

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

a(n) = Sum_{k} A330462(n,k) * k!.

A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 17, 32, 54, 100, 166, 289, 494, 840, 1393, 2400, 3931, 6498, 10861, 17728, 28863, 47557, 77042, 123881, 201172, 322459, 517032, 827993, 1316064, 2084632, 3328204, 5236828, 8247676, 13005652, 20417628, 31934709, 49970815, 77789059, 121144373

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 17 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(4)(), (31)(), (22)(), (211)(), (1111)(), (3)(1), (21)(1), (111)(1),
(2)(1)(), (11)(1)().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002865, A032305, A063834, A093637, A271619, A281113, A296118, A300383, A301422, A301462, A301467, A301480, A301706.

nn=50;
ser=x*Product[1+PartitionsP[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={Vec(prod(k=1, n-1, 1 + numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} (1 + A000041(n-1) x^n).

A304785 Expansion of Product_{k>=1} (1 - p(k)*x^k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, -1, -2, -1, -2, 4, 0, 15, 7, 17, 22, 26, -79, -2, -12, -392, -250, -392, -443, -640, -404, -795, 5106, 1147, 3304, 4542, 32330, 21001, 23372, 21015, 14496, 16165, -17213, 51296, -231330, -890169, -492310, -755449, -1648273, 131600, -6308274, -2160440, -4410945, 1593319

Offset: 0

Ilya Gutkovskiy, May 18 2018

sign

Convolution inverse of A063834.

Eric Weisstein's World of Mathematics, Partition Function P
Index entries for sequences related to partitions

Cf. A000041, A063834, A271619, A300508.

nmax = 43; CoefficientList[Series[Product[(1 - PartitionsP[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsP[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

G.f.: Product_{k>=1} (1 - A000041(k)*x^k).

A317535 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} 1/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 23, 48, 106, 227, 494, 1065, 2310, 4991, 10808, 23376, 50593, 109455, 236858, 512479, 1108924, 2399418, 5191853, 11233929, 24307777, 52596430, 113806948, 246252376, 532834797, 1152933975, 2494689316, 5397944266, 11679933875, 25272740480, 54684508281, 118324934647

Offset: 0

Ilya Gutkovskiy, Jul 30 2018

nonn

Invert transform of A000065.

Robert Israel, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A000041, A000065, A001970, A055887, A063834, A261049, A271619, A304966, A317536.

seq(coeff(series(1/(1+1/(1-x)-mul(1/(1-x^k),k=1..n)), x,n+1),x,n),n=0..40); # Muniru A Asiru, Jul 30 2018

nmax = 35; CoefficientList[Series[1/(1 + 1/(1 - x) - Product[1/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/(1 - Sum[(PartitionsP[k] - 1) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(PartitionsP[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 35}]

G.f.: 1/(1 - Sum_{k>=1} A000065(k)*x^k).

A336138 Number of set partitions of the binary indices of n with distinct block-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 5, 2, 4, 5, 12, 1, 2, 2, 5, 2, 5, 4, 13, 2, 4, 5, 13, 5, 13, 13, 43, 1, 2, 2, 5, 2, 5, 5, 13, 2, 5, 4, 14, 5, 13, 14, 42, 2, 4, 5, 13, 5, 14, 13, 43, 5, 13, 14, 45, 14, 44, 44, 160, 1, 2, 2, 5, 2, 5, 5, 14, 2, 5, 5, 14, 4, 13

Offset: 0

Gus Wiseman, Jul 12 2020

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(n) set partitions for n = 3, 7, 11, 15, 23:
  {12}    {123}      {124}      {1234}        {1235}
  {1}{2}  {1}{23}    {1}{24}    {1}{234}      {1}{235}
          {13}{2}    {12}{4}    {12}{34}      {12}{35}
          {1}{2}{3}  {14}{2}    {123}{4}      {123}{5}
                     {1}{2}{4}  {124}{3}      {125}{3}
                                {13}{24}      {13}{25}
                                {134}{2}      {135}{2}
                                {1}{2}{34}    {15}{23}
                                {1}{23}{4}    {1}{2}{35}
                                {1}{24}{3}    {1}{25}{3}
                                {14}{2}{3}    {13}{2}{5}
                                {1}{2}{3}{4}  {15}{2}{3}
                                              {1}{2}{3}{5}

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

The version for twice-partitions is A271619.
The version for partitions of partitions is (also) A271619.
These set partitions are counted by A275780.
The version for factorizations is A321469.
The version for normal multiset partitions is A326519.
The version for equal block-sums is A336137.
Set partitions with distinct block-lengths are A007837.
Set partitions of binary indices are A050315.
Twice-partitions with equal sums are A279787.
Partitions of partitions with equal sums are A305551.
Normal multiset partitions with equal block-lengths are A317583.
Multiset partitions with distinct block-sums are ranked by A326535.
Cf. A000110, A032011, A035470, A131632, A321455, A326026, A326514, A326517, A326518, A326534, A326565.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],UnsameQ@@Total/@#&]],{n,0,100}]

A301761 Number of ways to choose a rooted partition of each part in a constant rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 13, 12, 26, 31, 57, 43, 150, 78, 224, 293, 484, 232, 1190, 386, 2260, 2087, 2558, 1003, 11154, 4701, 7889, 13597, 30041, 3719, 83248, 5605, 95006, 84486, 63506, 251487, 654394, 17978, 169864, 490741, 2290336, 37339, 4079503, 53175, 3979370

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 13 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(2)(2), (2)(11), (11)(2), (11)(11),
(1)(1)(1),
()()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000041, A002865, A018818, A063834, A093637, A127524, A260685, A271619, A279787, A301422, A301462, A301467, A301480, A301706.

Table[Sum[PartitionsP[n/d-1]^d,{d,Divisors[n]}],{n,50}]

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

a(n) = Sum_{d | n-1} A000041((n-1)/d-1)^d for n > 1. - Andrew Howroyd, Aug 26 2018

A306320 Number of square plane partitions of n with distinct row sums and distinct column sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 5, 5, 10, 11, 18, 21, 31, 37, 56, 70, 97, 134, 180, 247, 343, 462, 623, 850, 1128, 1509, 2004, 2649, 3467, 4590, 5958, 7814, 10161, 13287, 17208, 22495, 29129, 37997, 49229, 64098, 82940, 107868, 139390, 180737, 233214, 301527, 388018, 500058

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

			The a(12) = 21 square plane partitions with distinct row sums and distinct column sums:
[twelve]
.
[64][73][82][91][54][63][72][81][44][53][53][62][62][71][43][43][52][52][61]
[11][11][11][11][21][21][21][21][31][22][31][22][31][31][32][41][32][41][41]
.
[221]
[211]
[111]

Cf. A000219, A089299 (square plane partitions), A101509, A271619, A279785, A306318, A323429, A323529, A323530, A323531.

Table[Sum[Length[Select[Union[Reverse/@Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],UnsameQ@@Total/@#&&UnsameQ@@Total/@If[#=={},{},Transpose[#]]&&And@@OrderedQ/@Reverse/@If[#=={},{},Transpose[#]]&]],{ptn,IntegerPartitions[n]}],{n,0,20}]

A316230 Expansion of Product_{k>=1} 1/(1 + p(k)*x^k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, -1, -1, -2, 1, -2, 2, 0, 24, -17, 31, -21, 94, -107, 121, -443, 742, -977, 532, -2159, 3275, -6193, 6988, -11156, 30278, -39214, 42759, -80255, 149070, -193093, 291229, -451125, 1017812, -1335848, 1609412, -3248202, 5606551, -7684574, 10012531, -17908468

Offset: 0

Ilya Gutkovskiy, Jun 27 2018

sign

Eric Weisstein's World of Mathematics, Partition Function P
Index entries for sequences related to partitions

Cf. A000041, A063834, A271619, A304784, A304785, A316231.

nmax = 39; CoefficientList[Series[Product[1/(1 + PartitionsP[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 39; CoefficientList[Series[Exp[Sum[Sum[(-1)^k PartitionsP[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-PartitionsP[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 39}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*p(j)^k*x^(j*k)/k).

cp's OEIS Frontend

A358826 Number of ways to choose a sequence of partitions, one of each part of an odd-length partition of 2n+1 into odd parts.

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A358827 Number of twice-partitions of n into partitions with all odd lengths and sums.

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A358913 Number of finite sequences of distinct sets with total sum n.

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A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

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A304785 Expansion of Product_{k>=1} (1 - p(k)*x^k), where p(k) = number of partitions of k (A000041).

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A317535 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} 1/(1 - x^k)).

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A336138 Number of set partitions of the binary indices of n with distinct block-sums.

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A301761 Number of ways to choose a rooted partition of each part in a constant rooted partition of n.

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