A358902
Number of integer compositions of n whose parts have weakly decreasing numbers of distinct prime factors (A001221).
Original entry on oeis.org
1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 84, 134, 213, 338, 536, 850, 1349, 2136, 3389, 5367, 8509, 13480, 21362, 33843, 53624, 84957, 134600, 213251, 337850, 535251, 847987, 1343440, 2128372, 3371895, 5341977, 8463051, 13407689, 21241181, 33651507, 53312538, 84460690
Offset: 0
The a(0) = 1 through a(6) = 13 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (21) (22) (23) (24)
(111) (31) (32) (33)
(211) (41) (42)
(1111) (221) (51)
(311) (222)
(2111) (231)
(11111) (321)
(411)
(2211)
(3111)
(21111)
(111111)
The strictly decreasing case is
A358903.
A116608 counts partitions by sum and number of distinct parts.
A334028 counts distinct parts in standard compositions.
A358836 counts multiset partitions with all distinct block sizes.
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p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
add((t-> `if`(t<=i, b(n-j, t), 0))(p(j)), j=1..n)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..40); # Alois P. Heinz, Feb 14 2024
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeNu/@#&]],{n,0,10}]
A358909
Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).
Original entry on oeis.org
1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306
Offset: 0
The complement is counted by
A358910.
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Table[Length[Select[IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]
A375405
Number of integer partitions of n with a repeated part other than the least.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 1, 3, 5, 8, 13, 20, 29, 42, 62, 83, 117, 158, 214, 283, 377, 488, 641, 823, 1058, 1345, 1714, 2154, 2713, 3387, 4222, 5230, 6474, 7959, 9782, 11956, 14591, 17737, 21529, 26026, 31422, 37811, 45425, 54418, 65097, 77652, 92510, 109943, 130468
Offset: 0
The a(0) = 0 through a(10) = 13 partitions:
. . . . . (221) (2211) (331) (332) (441) (442)
(2221) (3221) (3321) (3322)
(22111) (3311) (4221) (3331)
(22211) (22221) (4411)
(221111) (32211) (5221)
(33111) (32221)
(222111) (33211)
(2211111) (42211)
(222211)
(322111)
(331111)
(2221111)
(22111111)
The complement for maxima instead of minima is
A034296.
These partitions have ranks
A375397.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums
A374706.
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Table[Length[Select[IntegerPartitions[n], !SameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]
- or -
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@DeleteCases[#,Min@@#]&]],{n,0,30}]
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A_x(N) = {my(x='x+O('x^N), f=sum(i=1,N,sum(j=i+1,N-i, ((x^(i+(2*j)))/(1-x^i))*prod(k=i+1,N-i-(2*j), if(kJohn Tyler Rascoe, Aug 21 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.
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twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@Total/@#&&UnsameQ@@Length/@#&]],{n,0,10}]
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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.
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Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]
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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
A358910
Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 41, 56, 84, 113, 164, 218, 306, 401, 547, 711, 949, 1218, 1599, 2034, 2625, 3310, 4224, 5283, 6664, 8271, 10336, 12747, 15791, 19343, 23791, 28979, 35398, 42887, 52073, 62779, 75804, 90967, 109291, 130605
Offset: 0
The a(9) = 1 through a(14) = 11 partitions:
(54) (541) (74) (543) (76) (554)
(542) (741) (544) (743)
(5411) (5421) (742) (761)
(54111) (5422) (5432)
(5431) (5441)
(7411) (7421)
(54211) (54221)
(541111) (54311)
(74111)
(542111)
(5411111)
The complement is counted by
A358909.
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Table[Length[Select[IntegerPartitions[n],!GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]
A358837
Number of odd-length multiset partitions of integer partitions of n.
Original entry on oeis.org
0, 1, 2, 4, 7, 14, 28, 54, 106, 208, 399, 757, 1424, 2642, 4860, 8851, 15991, 28673, 51095, 90454, 159306, 279067, 486598, 844514, 1459625, 2512227, 4307409, 7357347, 12522304, 21238683, 35903463, 60497684, 101625958, 170202949, 284238857, 473356564, 786196353
Offset: 0
The a(1) = 1 through a(5) = 14 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}}
{{1},{1},{1,1}} {{1,1,1,1,1}}
{{1},{1},{3}}
{{1},{2},{2}}
{{1},{1},{1,2}}
{{1},{2},{1,1}}
{{1},{1},{1,1,1}}
{{1},{1,1},{1,1}}
{{1},{1},{1},{1},{1}}
The version for set partitions is
A024429.
These multiset partitions are ranked by
A026424.
The version for partitions is
A027193.
The version for twice-partitions is
A358824.
A001970 counts multiset partitions of integer partitions.
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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/@Reverse/@IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,10}]
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P(v,y) = {1/prod(k=1, #v, (1 - y*x^k + O(x*x^#v))^v[k])}
seq(n) = {my(v=vector(n, k, numbpart(k))); (Vec(P(v,1)) - Vec(P(v,-1)))/2} \\ Andrew Howroyd, Dec 31 2022
A350587
Expansion of Product_{k>=1} (1-x^k/Product_{j=1..k} (1-x^j)).
Original entry on oeis.org
1, -1, -2, -2, -2, 1, 4, 13, 22, 36, 47, 54, 37, -11, -129, -346, -709, -1257, -2023, -2979, -4014, -4836, -4851, -3041, 2310, 13785, 35186, 71598, 129624, 216732, 340488, 505769, 710775, 938823, 1146714, 1244936, 1070745, 347604, -1366923, -4751316
Offset: 0
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a[n_] := SeriesCoefficient[Product[1 - x^k/Product[1 - x^j, {j, 1, k}], {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 0, 39}] (* Robert P. P. McKone, Jan 18 2022 *)
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my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, 1-x^k/prod(j=1, k, 1-x^j)))
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