A319066 -id:A319066 - cp's OEIS frontend

A320322 Number of integer partitions of n whose product is a perfect power.

1, 0, 0, 0, 2, 2, 5, 5, 9, 11, 18, 19, 28, 30, 42, 50, 68, 76, 102, 113, 146, 170, 212, 241, 312, 356, 441, 514, 628, 720, 887, 1008, 1215, 1403, 1660, 1903, 2291, 2609, 3107, 3594, 4254, 4864, 5739, 6546, 7672, 8811, 10237, 11651, 13583, 15420, 17867, 20382

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

			The a(4) = 2 through a(11) = 19 integer partitions:
  4   41   33    331    8       9        55        551
  22  221  42    421    44      81       82        632
           222   2221   422     333      91        821
           411   4111   2222    441      433       911
           2211  22111  3311    4221     442       4331
                        4211    22221    811       4421
                        22211   33111    3322      8111
                        41111   42111    3331      33221
                        221111  222111   4222      33311
                                411111   4411      42221
                                2211111  22222     44111
                                         42211     222221
                                         222211    422111
                                         331111    2222111
                                         421111    3311111
                                         2221111   4211111
                                         4111111   22211111
                                         22111111  41111111
                                                   221111111

Cf. A003963, A064573, A305551, A319056, A319066, A319071, A319169, A320325.

Table[Length[Select[IntegerPartitions[n],GCD@@FactorInteger[Times@@#][[All,2]]>1&]],{n,30}]

A320324 Numbers of which each prime index has the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 85, 89, 91, 93, 97, 99, 101, 103, 107, 109, 113, 121, 123, 125, 127, 128, 131, 135, 137, 139, 149, 151, 153

Offset: 1

Gus Wiseman, Oct 10 2018

nonn

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

			The terms together with their corresponding multiset multisystems (A302242):
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}
  33: {{1},{3}}
  37: {{1,1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

Amiram Eldar, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A001222, A038041, A112798, A302242, A306017, A317583, A319066, A319169, A320325, A322794, A326533, A326534, A326535, A326536, A326537.

Select[Range[100],SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

is(n) = #Set(apply(p -> bigomega(primepi(p)), factor(n)[,1]~))<=1 \\ Rémy Sigrist, Oct 11 2018

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

A320325 Numbers whose product of prime indices is a perfect power.

Original entry on oeis.org

7, 9, 14, 18, 19, 21, 23, 25, 27, 28, 36, 38, 42, 46, 49, 50, 53, 54, 56, 57, 63, 72, 76, 81, 84, 92, 97, 98, 100, 103, 106, 108, 112, 114, 115, 121, 125, 126, 131, 133, 144, 147, 151, 152, 159, 161, 162, 168, 169, 171, 175, 183, 184, 185, 189, 194, 195, 196

Offset: 1

Gus Wiseman, Oct 10 2018

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 terms together with their corresponding multiset multisystems (A302242):
   7: {{1,1}}
   9: {{1},{1}}
  14: {{},{1,1}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  36: {{},{},{1},{1}}
  38: {{},{1,1,1}}
  42: {{},{1},{1,1}}
  46: {{},{2,2}}
  49: {{1,1},{1,1}}
  50: {{},{2},{2}}
  53: {{1,1,1,1}}
  54: {{},{1},{1},{1}}
  56: {{},{},{},{1,1}}
  57: {{1},{1,1,1}}
  63: {{1},{1},{1,1}}
  72: {{},{},{},{1},{1}}
  76: {{},{},{1,1,1}}
  81: {{1},{1},{1},{1}}

Cf. A000720, A001222, A001597, A003963, A056239, A064573, A112798, A302242, A305551, A306017, A319056, A319066, A320322, A320323, A320324.

Select[Range[100],GCD@@FactorInteger[Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]][[All,2]]>1&]

A101509 Binomial transform of tau(n) (see A000005).

Original entry on oeis.org

1, 3, 7, 16, 35, 75, 159, 334, 696, 1442, 2976, 6123, 12562, 25706, 52492, 107014, 217877, 443061, 899957, 1826078, 3701783, 7498261, 15178255, 30706320, 62085915, 125465715, 253415981, 511608490, 1032427637, 2082680887, 4199956101, 8467124805, 17064784905, 34382825363, 69256687719, 139465867773

Offset: 0

Paul Barry, Dec 05 2004

Row sums of A101508.

Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - M. F. Hasler, Jan 14 2009

			From _Gus Wiseman_, Jan 16 2019: (Start)
The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
  [4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
.
  [1] [3] [2] [1 1]
  [3] [1] [2] [1 1]
.
  [1] [1] [2]
  [1] [2] [1]
  [2] [1] [1]
.
  [1]
  [1]
  [1]
  [1]
(End)

M. F. Hasler, Table of n, a(n) for n = 0..500
L. Manor, M. F. Hasler, Partitions and A101509. SeqFan list, Jan 14 2009
N. J. A. Sloane, Transforms

Cf. A000005 (tau), A101508, A160399.

Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.

bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
a:= bintr(n-> numtheory[tau](n+1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2011

a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017 *)

A101509(n) = sum( k=0,n, numdiv(k+1)*binomial(n,k)) \\ M. F. Hasler, Jan 14 2009

a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - Joerg Arndt, Jan 30 2011
a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 07 2020

A319169 Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 7, 11, 11, 14, 15, 20, 19, 26, 27, 34, 35, 43, 45, 59, 60, 72, 77, 94, 98, 118, 125, 148, 158, 184, 198, 233, 245, 282, 308, 353, 374, 428, 464, 525, 566, 635, 686, 779, 832, 930, 1005, 1123, 1208, 1345, 1451, 1609, 1732, 1912

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

			The a(1) = 1 through a(9) = 6 integer partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    32     33      52       44        72
              1111  11111  222     322      53        333
                           111111  1111111  332       522
                                            2222      3222
                                            11111111  111111111

Alois P. Heinz, Table of n, a(n) for n = 0..2500 (first 101 terms from Chai Wah Wu)

Cf. A000607, A001222, A003963, A064573, A279787, A305551, A319056, A319066, A319071, A320322, A320324.

b:= proc(n, i, f) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, f)+(o-> `if`(f in {0, o}, b(n-i, min(i, n-i),
     `if`(f=0, o, f)), 0))(numtheory[bigomega](i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..75);  # Alois P. Heinz, Dec 15 2018

Table[Length[Select[IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,30}]
(* Second program: *)
b[n_, i_, f_] := b[n, i, f] = If[n == 0, 1, If[i < 1, 0,
     b[n, i-1, f] + Function[o, If[f == 0 || f == o, b[n-i, Min[i, n-i],
     If[f == 0, o, f]], 0]][PrimeOmega[i]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 75] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(51)-a(58) from Chai Wah Wu, Nov 12 2018

A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 25, 34, 54, 74, 109, 146, 211, 276, 381, 501, 675, 871, 1156, 1477, 1926, 2447, 3142, 3957, 5038, 6291, 7918, 9839, 12277, 15148, 18773, 23027, 28333, 34587, 42284, 51357, 62466, 75503, 91344, 109971, 132421, 158755, 190365, 227354, 271511

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(5) = 14 split partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

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

Cf. A000005, A000219, A008284, A101509, A316245, A319066, A323295, A323300, A323307, A323429, A323434.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, numtheory
      [tau](t+n), b(n, i-1, t)+b(n-i, min(n-i, i), t+1))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

Table[Sum[Length[Divisors[Length[ptn]]],{ptn,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1,
     DivisorSigma[0, t+n], b[n, i-1, t] + b[n-i, Min[n-i, i], t+1]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); Vec(1+sum(k=1, N, numdiv(k)*x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 21 2022

my(N=66, x='x+O('x^N)); Vec(1+sum(i=1, N, sum(j=1, N\i, x^(i*j)/prod(k=1, i*j, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022

a(n) = Sum_y A000005(k), where the sum is over all integer partitions of n and k is the number of parts.
From Seiichi Manyama, Jan 21 2022: (Start)
G.f.: 1 + Sum_{k>=1} A000005(k) * x^k/Product_{j=1..k} (1-x^j).
G.f.: 1 + Sum_{i>=1} Sum_{j>=1} x^(i*j)/Product_{k=1..i*j} (1-x^k). (End)
a(n) = Sum_{i=1..n} Sum_{j=1..n} A008284(n,i*j). - Ridouane Oudra, Apr 13 2023

A323429 Number of rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 58, 81, 124, 169, 257, 345, 501, 684, 968, 1304, 1830, 2452, 3387, 4541, 6188, 8257, 11193, 14865, 19968, 26481, 35341, 46674, 62007, 81611, 107860, 141602, 186292, 243800, 319610, 416984, 544601, 708690, 922472, 1197018, 1553442

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are weakly decreasing.

			The a(5) = 14 matrices:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323430, A323431, A323432, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A007713 Number of 4-level rooted trees with n leaves.

Original entry on oeis.org

1, 1, 4, 10, 30, 75, 206, 518, 1344, 3357, 8429, 20759, 51044, 123973, 299848, 719197, 1716563, 4070800, 9607797, 22555988, 52718749, 122655485, 284207304, 655894527, 1508046031, 3454808143, 7887768997, 17949709753, 40719611684, 92096461012, 207697731344

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, Oct 11 2018: (Start)
Also the number of multiset partitions of multiset partitions of integer partitions of n. For example, the a(1) = 1 through a(4) = 30 multiset partitions are:
  ((1))  ((2))       ((3))            ((4))
         ((11))      ((12))           ((13))
         ((1)(1))    ((111))          ((22))
         ((1))((1))  ((1)(2))         ((112))
                     ((1)(11))        ((1111))
                     ((1))((2))       ((1)(3))
                     ((1))((11))      ((2)(2))
                     ((1)(1)(1))      ((1)(12))
                     ((1))((1)(1))    ((2)(11))
                     ((1))((1))((1))  ((1)(111))
                                      ((11)(11))
                                      ((1))((3))
                                      ((2))((2))
                                      ((1))((12))
                                      ((1)(1)(2))
                                      ((2))((11))
                                      ((1))((111))
                                      ((1)(1)(11))
                                      ((11))((11))
                                      ((1))((1)(2))
                                      ((2))((1)(1))
                                      ((1))((1)(11))
                                      ((1)(1)(1)(1))
                                      ((11))((1)(1))
                                      ((1))((1))((2))
                                      ((1))((1))((11))
                                      ((1))((1)(1)(1))
                                      ((1)(1))((1)(1))
                                      ((1))((1))((1)(1))
                                      ((1))((1))((1))((1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Column k=4 of A290353.
Main diagonal of A055886.
Cf. A001970, A047968, A050342, A089259, A141268, A258466, A261049, A319066, A320328, A320330, A320331.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: b0:= etr(1): b1:= etr(b0): a:= etr(b1): seq(a(n), n=0..30); # Alois P. Heinz, Sep 08 2008

i[ n_, m_ ] := 1 /; m==1 || n==0; i[ n_, m_ ] := (i[ n, m ]=1/n Sum[ i[ k, m ] Plus @@ ((# i[ #, m-1 ])& /@ Divisors[ n-k ]), {k, 0, n-1} ]) /; n>0 && m>1
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b0 = etr[Function[1]]; b1 = etr[b0]; a = etr[b1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

Euler transform applied thrice to all-1's sequence.

A089299 Number of square plane partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 41, 57, 78, 108, 146, 202, 274, 375, 509, 690, 929, 1255, 1679, 2246, 2991, 3979, 5266, 6971, 9187, 12104, 15898, 20870, 27322, 35762, 46690, 60927, 79348, 103270, 134138, 174108, 225576, 291990, 377320, 487083

Offset: 0

N. J. A. Sloane, Dec 25 2003

nonn

Number of ways of writing n as a sum p(1,1) + p(1,2) + ... + p(1,k) + p(2,1) + ... + p(2,k) + ... + p(k,1) + ... + p(k,k) for some k so that in the square array {p(i,j)} the numbers are nonincreasing along rows and columns. All the p(i,j) are >= 1.

			a(7) = 5:
7 41 32 31 22
. 11 11 21 21
a(10) = 16 from {{10}}, {{3, 2}, {3, 2}}, {{3, 3}, {2, 2}}, {{3, 3}, {3, 1}}, {{4, 1}, {4, 1}}, {{4, 2}, {2, 2}}, {{4, 2}, {3, 1}}, {{4, 3}, {2, 1}}, {{4, 4}, {1, 1}}, {{5, 1}, {3, 1}}, {{5, 2}, {2, 1}}, {{5, 3}, {1, 1}}, {{6, 1}, {2, 1}}, {{6, 2}, {1, 1}}, {{7, 1}, {1, 1}}, {{2, 1, 1}, {1, 1, 1}, {1, 1, 1}}
From _Gus Wiseman_, Jan 16 2019: (Start)
The a(10) = 16 square plane partitions:
  [ten]
.
  [32] [33] [33] [41] [42] [42] [43] [44] [51] [52] [53] [61] [62] [71]
  [32] [22] [31] [41] [22] [31] [21] [11] [31] [21] [11] [21] [11] [11]
.
  [211]
  [111]
  [111]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A008763, A001970, A089292.

Cf. A000219, A003293, A101509, A319066, A323429, A323433, A323450.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 16 2019 *)

G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..2k-1} (1-x^j)^min(j,2k-j). - Franklin T. Adams-Watters, Jun 14 2006

Corrected and extended by Wouter Meeussen, Dec 30 2003
a(21)-a(25) from John W. Layman, Jan 02 2004
More terms from Franklin T. Adams-Watters, Jun 14 2006
Name edited by Gus Wiseman, Jan 16 2019

cp's OEIS Frontend

A320322 Number of integer partitions of n whose product is a perfect power.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A320324 Numbers of which each prime index has the same number of prime factors, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A320325 Numbers whose product of prime indices is a perfect power.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A101509 Binomial transform of tau(n) (see A000005).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A319169 Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A323429 Number of rectangular plane partitions of n.