A339006 -id:A339006 - cp's OEIS frontend

A133494 Diagonal of the array of iterated differences of A047848.

1, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883

Offset: 0

Paul Barry, Paul Curtz, Dec 23 2007

a(n) is the number of ways to choose a composition C, and then choose a composition of each part of C. - Geoffrey Critzer, Mar 19 2012
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the reptend length of 1/3^(n+1) in decimal. - Jianing Song, Nov 14 2018
Also the number of pairs of integer compositions, the first summing to n and the second with sum equal to the length of the first. If an integer composition is regarded as an arrow from sum to length, these are composable pairs, and the obvious composition operation founds a category of integer compositions. For example, we have (2,1,1,4) . (1,2,1) . (1,2) = (2,6), where dots represent the composition operation. The version without empty compositions is A000244. Composable triples are counted by 1 followed by A000302. The unordered version is A022811. - Gus Wiseman, Jul 14 2022

			From _Gus Wiseman_, Jul 15 2020: (Start)
The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
  ()  (1)  (2)      (3)
           (1,1)    (1,2)
           (1),(1)  (2,1)
                    (1,1,1)
                    (1),(2)
                    (2),(1)
                    (1),(1,1)
                    (1,1),(1)
                    (1),(1),(1)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (3).

The strict version is A336139.
Splittings of partitions are A323583.
Multiset partitions of partitions are A001970.
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 strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict compositions of each part of a composition are A307068.
Compositions of each part of a strict composition are A336127.
Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
Cf. A339006, A355382, A355384, A355387.

[n eq 0 select 1 else 3^(n-1): n in [0..30]]; // G. C. Greubel, Nov 20 2023

a:= n-> ceil(3^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2020

CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 3^(Range[0, 30])] (* G. C. Greubel, Nov 20 2023 *)

a(n)=max(1,3^(n-1)) \\ Charles R Greathouse IV, Jul 07 2011

Vec((1-2*x)/(1-3*x) + O(x^100)) \\  Altug Alkan, Oct 30 2015

[(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023

Binomial transform of A078008. - Paul Curtz, Aug 04 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: (1 - 2*x)/(1 - 3*x).
a(n) = A000244(n-1), n > 0. (End)
From Philippe Deléham, Nov 13 2008: (Start)
a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
Let A(x) be the g.f. Then B(x) = x*A(x) satisfies B(x/(1-x)) = x/(1 - 2*B(x)). - Vladimir Kruchinin, Dec 05 2011
G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - Joerg Arndt, Sep 30 2012
For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - J. Conrad, Oct 29 2015
G.f.: 1 + x/(1 + x)*(1 + 4*x/(1 + 4*x)*(1 + 7*x/(1 + 7*x)*(1 + 10*x/(1 + 10*x)*(1 + .... - Peter Bala, May 27 2017
Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - Gus Wiseman, Jul 19 2020
a(n) = ceiling(3^(n-1)). - Alois P. Heinz, Jul 26 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: (2 + exp(3*x))/3.
a(n) = 3*a(n-1) for n > 1. (End)

Definition clarified by R. J. Mathar, Nov 11 2008

A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 4, 12, 30, 66, 164, 419, 1049, 2625, 6372, 15451, 37335, 89855, 216523, 518714, 1235897, 2930050, 6911149, 16217817, 37914515, 88304358, 204971388, 474172899, 1093547574, 2513959446, 5761735383, 13165908506, 29998936859, 68164839887, 154478212575

Offset: 0

Gus Wiseman, Jul 01 2022

nonn

If a composition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of compositions.

			The initial terms count the following containments:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)
                (11)(1)  (21)(21)  (31)(31)
                         (12)(12)  (13)(13)
                         (111)(1)  (22)(2)
                                   (211)(11)
                                   (211)(21)
                                   (121)(11)
                                   (121)(12)
                                   (121)(21)
                                   (112)(11)
                                   (112)(12)
                                   (1111)(1)

Christian Sievers, Table of n, a(n) for n = 0..59

The homog. case is A032020, w/o containment A355388 (partitions A355385).
For partitions we have A355383, homog. A000009, w/ multiplicity A339006.
A000244 counts splittings of compositions, for partitions A323583.
Cf. A001970, A022811, A063834, A070933, A072706, A133494, A336139.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,5}]

a(21) and beyond from Christian Sievers, May 08 2025

A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 16, 26, 42, 64, 100, 150, 224, 330, 482, 697, 999, 1418, 1996, 2794, 3879, 5355, 7343, 10018, 13583, 18338, 24618, 32917, 43790, 58043, 76591, 100716, 131906, 172194, 223966, 290423, 375318, 483668, 621368, 796138, 1017146

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

If a partition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of partitions.

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(1)  (21)(21)  (31)(31)   (41)(41)
                         (111)(1)  (22)(2)    (32)(32)
                                   (211)(11)  (311)(11)
                                   (211)(21)  (311)(31)
                                   (1111)(1)  (221)(21)
                                              (221)(22)
                                              (2111)(11)
                                              (2111)(21)
                                              (11111)(1)

With multiplicity we have A339006.
The version for compositions is A355384.
The homogeneous version w/o containment is A355385, compositions A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Splitting partitions: A072706, A316245, A317715, A318683, A318684, A319794, A323433, A323583, A336131-A336136.
Cf. A000009, A022811, A032020, A070933, A181591, A181819, A319910, A355382.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,IntegerPartitions[n]}],{n,0,15}]

A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 25, 43, 81, 141, 243, 409, 699, 1132, 1844, 2995, 4744, 7408, 11655, 17839, 27509, 41546, 62879, 93537, 139974, 205547, 302714, 440097, 640968, 921774, 1327538, 1891548, 2696635, 3809860, 5380257, 7540778, 10561566, 14687109, 20408170, 28183998, 38882009

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

Also the number of composable pairs of integer partitions of n, where a partition is regarded as an arrow from (number of parts) to (number of distinct parts). Is there a nice choice of a composition operation making this into an associative category?

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(2)  (21)(21)  (31)(31)   (41)(41)
                         (111)(3)  (31)(22)   (41)(32)
                                   (22)(4)    (32)(41)
                                   (211)(31)  (32)(32)
                                   (211)(22)  (311)(41)
                                   (1111)(4)  (311)(32)
                                              (221)(41)
                                              (221)(32)
                                              (2111)(41)
                                              (2111)(32)
                                              (11111)(5)

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

The inhomogeneous version with containment and multiplicity is A339006.
The inhomogeneous version with containment is A355383.
The inhomogeneous version with containment for compositions is A355384.
The version for compositions is A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
A323583 counts splittings of partitions.
Cf. A000009, A008284, A022811, A032020, A070933, A072706, A116608, A279787, A319910, A336135.

Table[Length[Select[Tuples[IntegerPartitions[n],2],Length[Union[#[[1]]]]==Length[#[[2]]]&]],{n,0,15}]

\\ P gives A008284 and R gives A116608 as g.f.'s.
P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(n,y) = {prod(k=1, n, 1 + y/(1 - x^k) - y + O(x*x^n))}
seq(n) = {my(g=Vec(P(n,y)), h=Vec(R(n,y))); vector(n+1, i, my(p=g[i], q=h[i]); sum(j=0, poldegree(q), polcoef(p,j)*polcoef(q,j)))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{j >= 1} A116608(n,j) * A008284(n,j) for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A355382 Number of divisors d of n such that bigomega(d) = omega(n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 02 2022

nonn

The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
If positive integers are regarded as arrows from the number of prime factors to the number of distinct prime factors, this sequence counts divisible composable pairs. Is there a nice choice of a composition operation making this into an associative category?

			The set of divisors of 180 satisfying the condition is {12, 18, 20, 30, 45}, so a(180) = 5.

The version with multiplicity is A181591.
For partitions we have A355383, with multiplicity A339006.
The version for compositions is A355384.
Positions of first appearances are A355386.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 count prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A000712, A022811, A056239, A071625, A118914, A133494, A181819, A182850, A323014, A323022, A323023, A355385, A355388.

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,100}]

A339011 Sum over all partitions of n of the product of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 3, 8, 17, 34, 61, 107, 176, 284, 442, 676, 1007, 1483, 2140, 3055, 4299, 5993, 8255, 11284, 15272, 20529, 27373, 36274, 47735, 62484, 81293, 105251, 135555, 173818, 221836, 282003, 356980, 450256, 565765, 708537, 884296, 1100287, 1364736, 1687952, 2081724

Offset: 0

Alois P. Heinz, Nov 18 2020

nonn

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

Cf. A096541, A339006, A339312.

Essentially partial sums of A093694.

b:= proc(n, i, p, d) option remember; `if`(n=0, d*p, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n<=0 or i=0, [0$2],
     `if`(i=1, [1, n], b(n, i-1)+ (p-> p+[0, p[1]])(b(n-i, i))))
    end:
a:= proc(n) option remember; b(n$2)[2]+`if`(n<0, 0, a(n-1)) end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 25 2022

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, d*p, If[i < 1, 0,
     Sum[b[n - i*j, i - 1, p + j, If[j == 0, d, d + 1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

A339312 Sum over all partitions of n of the GCD of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 17, 23, 33, 47, 71, 92, 129, 169, 235, 299, 408, 525, 691, 885, 1147, 1427, 1832, 2312, 2878, 3635, 4519, 5631, 7002, 8637, 10514, 13055, 15864, 19396, 23530, 28702, 34746, 42210, 50671, 61224, 73506, 88394, 105447, 126398, 150588, 179075

Offset: 0

Alois P. Heinz, Dec 02 2020

nonn

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

Cf. A003989, A050873, A096541, A339006, A339011, A339394.

b:= proc(n, i, p, d) option remember; `if`(n=0, igcd(p, d),
      add(b(n-i*j, i-1, p+j, d+signum(j)), j=`if`(i>1, 0..n/i, n)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, GCD[p, d],
     Sum[b[n - i*j, i - 1, p + j, d + Sign[j]],
     {j, If[i > 1, Range[0, n/i], {n}]}]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

A355389 Number of unordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 1, 3, 10, 21, 55, 105, 231, 435, 861, 1540, 2926, 5050, 9045, 15400, 26565, 43956, 73920, 119805, 196251, 313236, 501501, 786885, 1239525, 1915903, 2965830, 4528545, 6909903, 10417330, 15699606, 23403061, 34848726, 51435153, 75761895, 110744403, 161577276

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

			The a(0) = 0 through a(4) = 10 pairs:
  .  .  (2)(11)  (3)(21)    (4)(22)
                 (3)(111)   (4)(31)
                 (21)(111)  (22)(31)
                            (4)(211)
                            (22)(211)
                            (31)(211)
                            (4)(1111)
                            (22)(1111)
                            (31)(1111)
                            (211)(1111)

The version for compositions is A006516.
Without distinctness we get A086737.
The unordered version is A355390, without distinctness A001255.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319794, A319910, A323433, A339006, A355385.

a:= n-> binomial(combinat[numbpart](n),2):
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 07 2024

Table[Binomial[PartitionsP[n],2],{n,0,6}]

a(n) = binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

a(n) = binomial(A000041(n), 2) = A355390(n)/2.

A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

Original entry on oeis.org

1, 12, 36, 120, 180, 360, 840, 1260, 5400, 27000, 2520, 5040, 6300, 7560, 15120, 12600, 25200

Offset: 1

Gus Wiseman, Jul 02 2022

The first position of -1 appears to be 18, pointed out by Amiram Eldar.
The terms are not always increasing.
The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.

			The terms together with their prime indices begin:
      1: {}
     12: {1,1,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    180: {1,1,2,2,3}
    360: {1,1,1,2,2,3}
    840: {1,1,1,2,3,4}
   1260: {1,1,2,2,3,4}
   5400: {1,1,1,2,2,2,3,3}
  27000: {1,1,1,2,2,2,3,3,3}
   2520: {1,1,1,2,2,3,4}
   5040: {1,1,1,1,2,2,3,4}
   6300: {1,1,2,2,3,3,4}
   7560: {1,1,1,2,2,2,3,4}
  15120: {1,1,1,1,2,2,2,3,4}
The terms together with their divisors satisfying the condition begin:
      1:   1
     12:   4,   6
     36:   4,   6,   9
    120:   8,  12,  20,  30
    180:  12,  18,  20,  30,  45
    360:   8,  12,  18,  20,  30,  45
    840:  24,  40,  56,  60,  84, 140, 210
   1260:  36,  60,  84,  90, 126, 140, 210, 315
   5400:   8,  12,  18,  20,  27,  30,  45,  50,  75
  27000:   8,  12,  18,  20,  27,  30,  45,  50,  75, 125
   2520:  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   5040:  16,  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   6300:  36,  60,  84,  90, 100, 126, 140, 150, 210, 225, 315, 350, 525

These are the positions of first appearances in A355382, which is the version of A181591 without multiplicity.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 counts prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
A355383 counts cmpsbl. pairs of partitions with containment, comps. A355384.
Cf. A000712, A022811, A056239, A071625, A181819, A319910, A339006.

tf=Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,1000}];
Table[Position[tf,n][[1,1]],{n,Select[Union[tf],SubsetQ[tf,Range[#]]&]}]

A108492 Sum of NumberOfParts!/NumberOfDifferentParts! for all integer partitions of n.

Original entry on oeis.org

1, 1, 3, 8, 31, 141, 819, 5562, 43773, 389203, 3858136, 42152116, 503098359, 6511429138, 90824834615, 1358130449902, 21672033893102, 367570633594883, 6602838664294634, 125228962373218571, 2500582942246200527, 52437202425839368049, 1152133477802718430790

Offset: 0

Thomas Wieder, Jun 05 2005

nonn

			a(4) = 31 because n=4 has the following A000041(4) = 5 partitions:
i=1: (1111) with p(1,4)=4, d(1,4)=1; 4!/1! = 24;
i=2: (112)  with p(2,4)=3, d(2,4)=2; 3!/2! =  3;
i=3: (13)   with p(3,4)=2, d(3,4)=2; 2!/2! =  1;
i=4: (4)    with p(4,4)=1, d(4,4)=1; 1!/1! =  1;
i=5: (22)   with p(5,4)=2, d(5,4)=1; 2!/1! =  2;
Their contributions sum up to 24+3+1+1+2 = 31 = a(4).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000041, A007318, A008277, A339006.

A108492 := proc(n::integer) local i,prttnlst,prttn,ZahlTeile,liste,ZahlVerschiedenerTeile,A108492;
# Procedure A108492 calculates the sequence A108492 for the integer partitions of n. prttn = an integer partition of n. See also http://www.thomas-wieder.privat.t-online.de/default.html
prttnlst:=partition(n); A108492 := 0; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; ZahlTeile := nops(prttn); liste := convert(prttn,multiset); ZahlVerschiedenerTeile := nops(liste); A108492 := A108492 + (ZahlTeile!/ZahlVerschiedenerTeile!); od; print(n,A108492); end proc;
# Second Maple program:
b:= proc(n,i,l,p,d) option remember;
      if n<0 then 0
    elif n=0 then p!/d!
    elif i=0 then 0
    else b(n, i-1, l, p, d) +b(n-i, i, i, p+1, `if`(i=l, d, d+1))
      fi
    end:
a:= n-> b(n, n, 0, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 25 2011

b[n_, i_, l_, p_, d_] := b[n, i, l, p, d] = Which[n<0, 0, n==0, p!/d!, i==0, 0, True, b[n, i-1, l, p, d]+b[n-i, i, i, p+1, If[i==l, d, d+1]]];
a[n_] := b[n, n, 0, 0, 0];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 18 2020, after Alois P. Heinz *)

a(n) = Sum_{i=1..A000041(n)} p(i,n)!/d(i,n)! with p(i,n) = number of parts of the i-th partition of n and d(i,n) = number of different parts of the i-th partition of n.

a(n) ~ n! * (1 + 1/(2*n) + 1/n^2 + 13/(6*n^3) + 19/(3*n^4) + 22/n^5 + 2057/(24*n^6)). - Vaclav Kotesovec, Sep 06 2015

cp's OEIS Frontend

A133494 Diagonal of the array of iterated differences of A047848.

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A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

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Mathematica

Extensions

A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

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Mathematica

A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.

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Mathematica

PARI

Formula

Extensions

A355382 Number of divisors d of n such that bigomega(d) = omega(n).

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Mathematica

A339011 Sum over all partitions of n of the product of the number of parts and the number of distinct parts.

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Maple

Mathematica

A339312 Sum over all partitions of n of the GCD of the number of parts and the number of distinct parts.

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Maple

Mathematica

A355389 Number of unordered pairs of distinct integer partitions of n.

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