A063834
Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.
Original entry on oeis.org
1, 1, 3, 6, 15, 28, 66, 122, 266, 503, 1027, 1913, 3874, 7099, 13799, 25501, 48508, 88295, 165942, 299649, 554545, 997281, 1817984, 3245430, 5875438, 10410768, 18635587, 32885735, 58399350, 102381103, 180634057, 314957425, 551857780, 958031826, 1667918758
Offset: 0
G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + 266*x^8 + ...
If n=6, a possible first partitioning is (3+3), resulting in the following second partitionings: ((3),(3)), ((3),(2+1)), ((3),(1+1+1)), ((2+1),(3)), ((2+1),(2+1)), ((2+1),(1+1+1)), ((1+1+1),(3)), ((1+1+1),(2+1)), ((1+1+1),(1+1+1)).
A001970 counts multiset partitions of integer partitions.
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1)+`if`(i>n, 0, numbpart(i)*b(n-i, i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Nov 26 2015
-
Table[Plus @@ Apply[Times, IntegerPartitions[i] /. i_Integer :> PartitionsP[i], 2], {i, 36}]
(* second program: *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i > n, 0, PartitionsP[i]*b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
-
{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 19 2016 */
A375133
Number of integer partitions of n whose maximal anti-runs have distinct maxima.
Original entry on oeis.org
1, 1, 1, 2, 3, 4, 5, 8, 10, 14, 17, 23, 29, 38, 47, 60, 74, 93, 113, 141, 171, 211, 253, 309, 370, 447, 532, 639, 758, 904, 1066, 1265, 1487, 1754, 2053, 2411, 2813, 3289, 3823, 4454, 5161, 5990, 6920, 8005, 9223, 10634, 12218, 14048, 16101, 18462, 21107
Offset: 0
The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with maxima (6,5,3), so y is counted under a(29).
The a(0) = 1 through a(9) = 14 partitions:
() (1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (31) (32) (42) (43) (53) (54)
(211) (41) (51) (52) (62) (63)
(311) (321) (61) (71) (72)
(411) (322) (422) (81)
(421) (431) (432)
(511) (521) (522)
(3211) (611) (531)
(3221) (621)
(4211) (711)
(4221)
(4311)
(5211)
(32211)
Includes all strict partitions
A000009.
For compositions instead of partitions we have
A374761.
The complement for minima instead of maxima is
A375404, ranks
A375399.
A011782 counts integer compositions.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums
A374706.
Cf.
A141199,
A279790,
A358830,
A358833,
A358836,
A358905,
A374704,
A374757,
A374758,
A375136,
A375400.
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]
-
A_x(N) = {my(x='x+O('x^N), f=sum(i=0,N,(x^i)*prod(j=1,i-1,(1-x^(3*j))/(1-x^j)))); Vec(f)}
A_x(51) \\ John Tyler Rascoe, Aug 21 2024
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
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))
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 plane partitions is
A323429.
The case of constant sums also is
A358833.
A055887 counts sequences of partitions with total sum n.
-
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
A374704
Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.
Original entry on oeis.org
1, 1, 3, 6, 15, 31, 77, 171, 410, 957, 2275, 5370, 12795, 30366, 72307, 172071, 409875, 976155, 2325804, 5541230, 13204161, 31464226, 74980838, 178684715, 425830008, 1014816979, 2418489344, 5763712776, 13736075563, 32735874251, 78016456122, 185929792353, 443110675075
Offset: 0
The a(0) = 1 through a(4) = 15 ways:
() ((1)) ((2)) ((3)) ((4))
((1,1)) ((1,2)) ((1,3))
((1),(1)) ((1,1,1)) ((2,2))
((1),(1,1)) ((1,1,2))
((1,1),(1)) ((2),(2))
((1),(1),(1)) ((1,1,1,1))
((1),(1,2))
((1,2),(1))
((1),(1,1,1))
((1,1),(1,1))
((1,1,1),(1))
((1),(1),(1,1))
((1),(1,1),(1))
((1,1),(1),(1))
((1),(1),(1),(1))
A variation for weakly increasing lengths is
A141199.
For identical sums instead of minima we have
A279787.
For maxima instead of minima, or for unreversed partitions, we have
A358905.
A055887 counts sequences of partitions with total sum n.
Cf.
A000041,
A063834,
A106356,
A189076,
A238343,
A304969,
A305551,
A319066,
A323429,
A333213,
A358833,
A358835.
-
Table[Length[Select[Join@@Table[Tuples[IntegerPartitions/@y], {y,Join@@Permutations/@IntegerPartitions[n]}],SameQ@@Min/@#&]],{n,0,15}]
-
seq(n) = Vec(1 + sum(k=1, n, -1 + 1/(1 - x^k/prod(j=k, n-k, 1 - x^j, 1 + O(x^(n-k+1)))))) \\ Andrew Howroyd, Dec 29 2024
A358832
Number of twice-partitions of n into partitions of distinct lengths and distinct sums.
Original entry on oeis.org
1, 1, 2, 4, 7, 15, 25, 49, 79, 154, 248, 453, 748, 1305, 2125, 3702, 5931, 9990, 16415, 26844, 43246, 70947, 113653, 182314, 292897, 464614, 739640, 1169981, 1844511, 2888427, 4562850, 7079798, 11064182, 17158151, 26676385, 41075556, 63598025, 97420873, 150043132
Offset: 0
The a(1) = 1 through a(5) = 15 twice-partitions:
(1) (2) (3) (4) (5)
(11) (21) (22) (32)
(111) (31) (41)
(11)(1) (211) (221)
(1111) (311)
(21)(1) (2111)
(111)(1) (11111)
(21)(2)
(22)(1)
(3)(11)
(31)(1)
(111)(2)
(211)(1)
(111)(11)
(1111)(1)
This is the case of
A271619 with distinct lengths.
This is the case of
A358830 with distinct sums.
For constant instead of distinct lengths and sums we have
A358833.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@Total/@#&&UnsameQ@@Length/@#&]],{n,0,10}]
-
seq(n)={ local(Cache=Map());
my(g=Vec(-1+1/prod(k=1, n, 1 - y*x^k + O(x*x^n))));
my(F(m,r,b) = my(key=[m,r,b], z); if(!mapisdefined(Cache,key,&z),
z = if(r<=0||m==0, r==0, self()(m-1, r, b) + sum(k=1, m, my(c=polcoef(g[m],k)); if(!bittest(b,k)&&c, c*self()(min(m-1,r-m), r-m, bitor(b, 1<Andrew Howroyd, Dec 31 2022
A358835
Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.
Original entry on oeis.org
1, 1, 3, 4, 8, 8, 17, 16, 31, 34, 54, 57, 108, 102, 166, 191, 294, 298, 504, 491, 803, 843, 1251, 1256, 2167, 1974, 3133, 3226, 4972, 4566, 8018, 6843, 11657, 11044, 17217, 15010, 28422, 21638, 38397, 35067, 58508, 44584, 91870, 63262, 125114, 106264, 177483
Offset: 0
The a(1) = 1 through a(6) = 17 multiset partitions:
{1} {2} {3} {4} {5} {6}
{11} {12} {13} {14} {15}
{1}{1} {111} {22} {23} {24}
{1}{1}{1} {112} {113} {33}
{1111} {122} {114}
{2}{2} {1112} {123}
{11}{11} {11111} {222}
{1}{1}{1}{1} {1}{1}{1}{1}{1} {1113}
{1122}
{3}{3}
{11112}
{111111}
{12}{12}
{2}{2}{2}
{111}{111}
{11}{11}{11}
{1}{1}{1}{1}{1}{1}
The version for set partitions is
A327899.
For distinct instead of constant lengths and sums we have
A358832.
The version for twice-partitions is
A358833.
A001970 counts multiset partitions of integer partitions.
-
Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]
-
P(n,y) = 1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022
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