A001255 -id:A001255 - cp's OEIS frontend

A054440 Number of ordered pairs of partitions of n with no common parts.

1, 0, 2, 4, 12, 16, 48, 60, 148, 220, 438, 618, 1302, 1740, 3216, 4788, 8170, 11512, 19862, 27570, 45448, 64600, 100808, 141724, 223080, 307512, 465736, 652518, 968180, 1334030, 1972164, 2691132, 3902432, 5347176, 7611484, 10358426, 14697028, 19790508, 27691500

Offset: 0

Herbert S. Wilf, May 13 2000

			a(3)=4 because of the 4 pairs of partitions of 3: (3,21),(3,111),(21,3),(111,3).

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Cf. A000041, A001255, A001318, A087960, A260672, A260664, A260669, A304873, A304877.

Main diagonal of A284592.

a054440 = sum . zipWith (*) a087960_list . map a001255 . a260672_row
-- Reinhard Zumkeller, Nov 15 2015

with(combinat): p1 := sum(numbpart(n)^2*x^n, n=0..500): it := p1*product((1-x^i), i=1..500): s := series(it, x, 500): for i from 0 to 100 do printf(`%d,`,coeff(s,x,i)) od:

nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^2*x^k, {k, 0, nmax}]/Sum[PartitionsP[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)

G.f.: Sum[p(n)^2*x^n]/Sum[p(n)*x^n], with p(n)=number of partitions of n.
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n)) / (64 * 2^(1/4) * n^(7/4)). - Vaclav Kotesovec, May 20 2018
a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k / (1 - x^k) + y^k / (1 - y^k)). - Ilya Gutkovskiy, Apr 24 2025

Corrected and extended by James Sellers, May 23 2000

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204

Offset: 0

Wouter Meeussen, Jul 09 2005

nonn

Counted as orderless pairs since intersection is commutative.

			Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
From _Gus Wiseman_, Oct 07 2023: (Start)
The a(6) = 4 pairs are:
  ((6),(5,1))
  ((6),(4,2))
  ((6),(3,2,1))
  ((5,1),(4,2))
(End)

Vaclav Kotesovec Table of n, a(n) for n = 0..1050 (terms 0..700 from Alois P. Heinz)

Column k=2 of A258280.
Cf. A079122, A086737.
Main diagonal of A284593 times (1/2).
This is the strict case of A260669.
The ordered version is A365662 = strict case of A054440.
This is the disjoint case of A366132, with twins A366317.
A000041 counts integer partitions, strict A000009.
A002219 counts biquanimous partitions, strict A237258, ordered A064914.
Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.

using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* Gus Wiseman, Oct 07 2023 *)

a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017

a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

Name edited by Gus Wiseman, Oct 10 2023

a(0)=1 prepended by Alois P. Heinz, Feb 09 2024

A365662 Number of ordered pairs of disjoint strict integer partitions of n.

Original entry on oeis.org

1, 0, 0, 2, 2, 6, 8, 14, 18, 32, 42, 66, 92, 136, 190, 280, 374, 532, 744, 1014, 1366, 1896, 2512, 3384, 4526, 6006, 7910, 10496, 13648, 17842, 23338, 30116, 38826, 50256, 64298, 82258, 105156, 133480, 169392, 214778, 270620, 340554, 428772, 536302, 670522

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of ways to first choose a strict partition of 2n, then a subset of it summing to n.

			The a(0) = 1 through a(7) = 14 pairs:
  ()()  .  .  (21)(3)  (31)(4)  (32)(5)   (42)(6)   (43)(7)
              (3)(21)  (4)(31)  (41)(5)   (51)(6)   (52)(7)
                                (5)(32)   (6)(42)   (61)(7)
                                (5)(41)   (6)(51)   (7)(43)
                                (32)(41)  (321)(6)  (7)(52)
                                (41)(32)  (42)(51)  (7)(61)
                                          (51)(42)  (421)(7)
                                          (6)(321)  (43)(52)
                                                    (43)(61)
                                                    (52)(43)
                                                    (52)(61)
                                                    (61)(43)
                                                    (61)(52)
                                                    (7)(421)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

For subsets instead of partitions we have A000244, non-disjoint A000302.
If the partitions can have different sums we get A032302.
The non-strict version is A054440, non-disjoint A001255.
The unordered version is A108796, non-strict A260669.
A000041 counts integer partitions, strict A000009.
A000124 counts distinct possible sums of subsets of {1..n}.
A000712 counts distinct submultisets of partitions.
A002219 and A237258 count partitions of 2n including a partition of n.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A006827, A046663, A064914, A122768, A260664, A365661, A365663.

Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2], Intersection@@#=={}&]], {n,0,15}]
Table[SeriesCoefficient[Product[(1 + x^k + y^k), {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Apr 24 2025 *)

a(n) = 2*A108796(n) for n > 1.

a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k). - Ilya Gutkovskiy, Apr 24 2025

A006907 Number of zeros in character table of symmetric group S_n.

Original entry on oeis.org

0, 0, 1, 4, 10, 29, 55, 153, 307, 588, 1018, 2230, 3543, 6878, 11216, 20615, 33355, 57980, 90194, 155176, 239327, 395473, 604113, 970294, 1453749, 2323476, 3425849, 5349414, 7905133, 11963861, 17521274, 26472001, 38054619, 56756488, 81683457, 119005220, 170498286, 247619748

Offset: 1

Simon Plouffe and N. J. A. Sloane

J. McKay, personal communication to N. J. A. Sloane, circa 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alexander R. Miller, On parity and characters of symmetric groups, preprint.
Alexander R. Miller, On parity and characters of symmetric groups, J. Combin. Theory Ser. A 162 (2019), 231-240.

Cf. A006908, A051748, A051749.

A006907 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Number(chi, x->x=0)); # Eric M. Schmidt, Jul 13 2012, simplified Jul 26 2012

a[n_] := Count[FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"], 0, 2]; Array[a, 10] (* Jean-François Alcover, Oct 08 2016 *)

a(n) + A051748(n) + A051749(n) = A001255(n). - R. J. Mathar, Mar 09 2022

More terms from Vladeta Jovovic, May 20 2003
More terms from Eric M. Schmidt, Jul 13 2012
a(36)-a(38) found by Alexander R. Miller (see 2019 reference). - N. J. A. Sloane, Jul 07 2020

A133042 Cubes of partition numbers.

Original entry on oeis.org

1, 1, 8, 27, 125, 343, 1331, 3375, 10648, 27000, 74088, 175616, 456533, 1030301, 2460375, 5451776, 12326391, 26198073, 57066625, 117649000, 246491883, 496793088, 1006012008, 1976656375, 3906984375, 7506509912, 14455457856

Offset: 0

Omar E. Pol, Oct 30 2007

nonn

			a(10) = 74088 because the partition number of 10 is 42 and 42^3 is 74088.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000

Cf. A000578, A030078. Partition number: A000041.

Cf. A001255, A260664.

a133042 = (^ 3) . a000041  -- Reinhard Zumkeller, Nov 15 2015

Table[PartitionsP[n]^3, {n, 0, 40}] (* Vaclav Kotesovec, Dec 01 2015 *)

for(n=0,20, print1(numbpart(n)^3, ", ")) \\ G. C. Greubel, Oct 02 2017

a(n) = A000041(n)^3.

a(n) ~ exp(Pi*sqrt(6*n)) / (192*sqrt(3)*n^3). - Vaclav Kotesovec, Dec 01 2015

A259399 a(n) = Sum_{k=0..n} p(k)^2, where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 6, 15, 40, 89, 210, 435, 919, 1819, 3583, 6719, 12648, 22849, 41074, 72050, 125411, 213620, 361845, 601945, 995074, 1622338, 2626342, 4201367, 6681992, 10515756, 16449852, 25509952, 39333476, 60172701, 91577517, 138390481, 208096282, 310976731, 462512831

Offset: 0

Vaclav Kotesovec, Jun 26 2015

nonn

In general, Sum_{k=0..n} p(k)^m ~ sqrt(6*n)/(m*Pi) * p(n)^m ~ exp(m*Pi*sqrt(2*n/3)) / (m * Pi * 3^((m-1)/2) * 2^(2*m-1/2) * n^(m-1/2)), for m >= 1.

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

Cf. A000041, A000070, A209536, A265093.

Partial sums of A001255.

a:= proc(n) option remember; `if`(n<0, 0,
      combinat[numbpart](n)^2+a(n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 21 2018

Table[Sum[PartitionsP[k]^2,{k,0,n}],{n,0,50}]

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (16*sqrt(6)*Pi*n^(3/2)).

a(n) = 1 + A209536(n). - Alois P. Heinz, Oct 21 2018

A304873 G.f.: Sum_{k>=0} p(k)^4 * x^k / Sum_{k>=0} p(k)*x^k, where p(n) is the partition function A000041(n).

Original entry on oeis.org

1, 0, 14, 64, 528, 1696, 11616, 33600, 169072, 525760, 2069922, 5928066, 22259874, 59321760, 193797792, 526647420, 1566376990, 4012181104, 11456306798, 28263784110, 75995086336, 184440427360, 468750673616, 1104027571108, 2730165482640, 6239956155696

Offset: 0

Vaclav Kotesovec, May 20 2018

nonn

In general, if m > 1 and g.f. = Sum_{k>=0} p(k)^m * x^k / Sum_{k>=0} p(k)*x^k, then a(n, m) ~ exp(Pi*sqrt(2*(m^2 - 1)*n/3)) * ((m^2 - 1)^(m - 3/4) / (2^(2*m - 3/4) * 3^(m/2 - 1/4) * m^(2*m - 1) * n^(m - 1/4))).

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A054440 (m=2), A260664 (m=3).

Cf. A000041, A001255, A133042, A304877, A304878.

nmax = 25; CoefficientList[Series[Sum[PartitionsP[k]^4*x^k, {k, 0, nmax}] / Sum[PartitionsP[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(3/4) * 3^(3/2) * 5^(13/4) * exp(Pi*sqrt(10*n)) / (2^22 * n^(15/4)).

A304990 Squares of number of partitions into distinct parts.

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 16, 25, 36, 64, 100, 144, 225, 324, 484, 729, 1024, 1444, 2116, 2916, 4096, 5776, 7921, 10816, 14884, 20164, 27225, 36864, 49284, 65536, 87616, 115600, 152100, 200704, 262144, 342225, 446224, 577600, 746496, 964324, 1238769, 1587600

Offset: 0

Vaclav Kotesovec, May 23 2018

nonn

Cf. A000009, A000290, A001255.

Mathematica
```
Table[PartitionsQ[n]^2, {n, 0, 50}]
```

a(n) ~ exp(2*Pi*sqrt(n/3)) / (16*sqrt(3)*n^(3/2)).
a(n) = A000009(n)^2.
a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k) * (1 + y^k). - Ilya Gutkovskiy, Apr 24 2025

A182988 The number of dominance pairs of integer partitions of n according to either/or dominance order, where dominance between two partitions x and y means that x is majorized by y or y is majorized by x.

Original entry on oeis.org

1, 1, 4, 9, 25, 49, 117, 217, 454, 830, 1594, 2796, 5159, 8777, 15415, 25810, 43819, 71595, 118629, 190148, 307519, 485660, 769382, 1195807, 1864617, 2857630, 4384962, 6641332, 10052272, 15043925, 22501510, 33315580, 49267369, 72250341, 105746966, 153646123

Offset: 0

Stephen DeSalvo, Feb 06 2011, Feb 13 2011

nonn

For two integer partitions of n chosen uniformly at random, a(n)/p(n)^2, where p(n) is the number of partitions of n, is the probability that one dominates the other.
As an example, consider the partitions (4,3,1) and (3,3,2).
4 >= 3, 4+3 >= 3+3, and 4+3+1 = 3+3+2, so we say (4,3,1) majorizes/dominates (3,3,2).
As a non-example, consider (4,1,1,1) and (3,3,1).
4 >= 3, but 4+1 < 3+3, so (4,1,1,1) does NOT dominate (3,3,1).
3 < 4, so (3,3,1) does NOT dominate (4,1,1,1).
Thus the pair (4,1,1,1) and (3,3,1) is not a dominance pair, and does not contribute to a(7).

			For n=1,2,3,4,5, a(n) = p(n)^2, since these values of n give a linear order for integer partitions.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n=1..55 from Stephen DeSalvo)
Wikipedia, Dominance Order
Wikipedia, Majorization

Cf. A000041, A001255, A248476, A265506.

b:= proc(n, m, i, j, t) option remember; `if`(n0,
       b(n, m, i, j-1, true), 0)+b(n, m, i-1, j, false)+
       b(n-i, m-j, min(n-i,i), min(m-j,j), true))))
    end:
a:= n-> 2*b(n$4, true)-combinat[numbpart](n):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 09 2015

b[n_, m_, i_, j_, t_] := b[n, m, i, j, t] = If[n0, b[n, m, i, j-1, True], 0] + b[n, m, i-1, j, False] + b[n-i, m-j, Min[n-i, i], Min[m-j, j], True]]]]; a[n_] := 2*b[n, n, n, n, True] - PartitionsP[n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Dec 09 2016 after Alois P. Heinz *)

a(0)=1 prepended by Alois P. Heinz, Jul 07 2015

A051749 Number of character table entries of the symmetric group S_n which are > 0.

Original entry on oeis.org

1, 3, 6, 14, 26, 58, 98, 194, 344, 652, 1165, 2020, 3552, 6077, 10362, 17080, 28570, 46836, 77045, 122013, 198461, 310602, 494008, 767237, 1205391, 1828252, 2846995, 4277605, 6520106, 9795470, 14738493, 21750402, 32582580, 47614253, 70213289

Offset: 1

JOHN MCKAY (mckay(AT)cs.concordia.ca), Dec 08 1999

Cf. A006907, A051748.

A051749 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Number(chi, x->x>0)); # Eric M. Schmidt, Jul 14 2012

a[n_] := Count[FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"], ?Positive, 2]; Array[a, 10] (* _Jean-François Alcover, Apr 03 2017 *)

A006907(n) + A051748(n) + a(n) = A001255(n). - R. J. Mathar, Mar 09 2022

More terms from Eric M. Schmidt, Jul 14 2012

cp's OEIS Frontend

A054440 Number of ordered pairs of partitions of n with no common parts.

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A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

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A365662 Number of ordered pairs of disjoint strict integer partitions of n.

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A006907 Number of zeros in character table of symmetric group S_n.

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A133042 Cubes of partition numbers.

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A259399 a(n) = Sum_{k=0..n} p(k)^2, where p(k) is the partition function A000041.

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A304873 G.f.: Sum_{k>=0} p(k)^4 * x^k / Sum_{k>=0} p(k)*x^k, where p(n) is the partition function A000041(n).

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