A358912
Number of finite sequences of integer partitions with total sum n and all distinct lengths.
Original entry on oeis.org
1, 1, 2, 5, 11, 23, 49, 103, 214, 434, 874, 1738, 3443, 6765, 13193, 25512, 48957, 93267, 176595, 332550, 622957, 1161230, 2153710, 3974809, 7299707, 13343290, 24280924, 43999100, 79412942, 142792535, 255826836, 456735456, 812627069, 1440971069, 2546729830
Offset: 0
The a(1) = 1 through a(4) = 11 sequences:
(1) (2) (3) (4)
(11) (21) (22)
(111) (31)
(1)(11) (211)
(11)(1) (1111)
(11)(2)
(1)(21)
(2)(11)
(21)(1)
(1)(111)
(111)(1)
The case of set partitions is
A007837.
This is the case of
A055887 with all distinct lengths.
For distinct sums instead of lengths we have
A336342.
The case of twice-partitions is
A358830.
The version for constant instead of distinct lengths is
A358905.
A141199 counts sequences of partitions with weakly decreasing lengths.
Cf.
A000219,
A001970,
A038041,
A060642,
A218482,
A271619,
A319066,
A358831,
A358901,
A358906,
A358908.
-
ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@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)); [subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, n, 1 + y*polcoef(g, k, y) + O(x*x^n)))]} \\ Andrew Howroyd, Dec 30 2022
A327607
Number of parts in all twice partitions of n where the first partition is strict.
Original entry on oeis.org
0, 1, 3, 11, 21, 58, 128, 276, 516, 1169, 2227, 4324, 8335, 15574, 29116, 55048, 97698, 176291, 323277, 563453, 1005089, 1770789, 3076868, 5293907, 9184885, 15668638, 26751095, 45517048, 76882920, 128738414, 217219751, 360525590, 599158211, 995474365
Offset: 0
a(3) = 11 = 1+2+3+2+3 counting the parts in 3, 21, 111, 2|1, 11|1.
-
g:= proc(n) option remember; (p-> [p(n), add(p(n-j)*
numtheory[tau](j), j=1..n)])(combinat[numbpart])
end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 (f-> f+[0, f[1]*
h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
-
g[n_] := g[n] = {PartitionsP[n], Sum[PartitionsP[n - j] DivisorSigma[0, j], {j, 1, n}]};
b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, Module[{h, f}, h = g[i]; f = b[n - i, Min[n - i, i - 1]] h[[1]]; b[n, i - 1] + f + {0, f[[1]] h[[2]] / h[[1]]}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)
A336141
Number of ways to choose a strict composition of each part of an integer partition of n.
Original entry on oeis.org
1, 1, 2, 5, 9, 17, 41, 71, 138, 270, 518, 938, 1863, 3323, 6163, 11436, 20883, 37413, 69257, 122784, 221873, 397258, 708142, 1249955, 2236499, 3917628, 6909676, 12130972, 21251742, 36973609, 64788378, 112103360, 194628113, 336713377, 581527210, 1000153063
Offset: 0
The a(1) = 1 through a(5) = 17 ways:
(1) (2) (3) (4) (5)
(1),(1) (1,2) (1,3) (1,4)
(2,1) (3,1) (2,3)
(2),(1) (2),(2) (3,2)
(1),(1),(1) (3),(1) (4,1)
(1,2),(1) (3),(2)
(2,1),(1) (4),(1)
(2),(1),(1) (1,2),(2)
(1),(1),(1),(1) (1,3),(1)
(2,1),(2)
(3,1),(1)
(2),(2),(1)
(3),(1),(1)
(1,2),(1),(1)
(2,1),(1),(1)
(2),(1),(1),(1)
(1),(1),(1),(1),(1)
Multiset partitions of partitions are
A001970.
Splittings of partitions are
A323583.
Splittings of partitions with distinct sums are
A336131.
Partitions:
- 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 partition are
A270995.
- Strict compositions of each part of a partition are
A336141.
Strict partitions:
- Partitions of each part of a strict partition are
A271619.
- Compositions of each part of a strict partition are
A304961.
- Strict partitions of each part of a strict partition are
A279785.
- Strict compositions of each part of a strict partition are
A336142.
Compositions:
- Partitions of each part of a composition are
A055887.
- Compositions of each part of a composition are
A133494.
- Strict partitions of each part of a composition are
A304969.
- Strict compositions of each part of a composition are
A307068.
Strict compositions:
- Partitions of each part of a strict composition are
A336342.
- Compositions of each part of a strict composition are
A336127.
- Strict partitions of each part of a strict composition are
A336343.
- Strict compositions of each part of a strict composition are
A336139.
-
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38); # Alois P. Heinz, Jul 31 2020
-
Table[Length[Join@@Table[Tuples[Join@@Permutations/@Select[IntegerPartitions[#],UnsameQ@@#&]&/@ctn],{ctn,IntegerPartitions[n]}]],{n,0,10}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
If[n==0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[n==0 || i==1, 1, g[n, i-1] +
b[i, i, 0] g[n-i, Min[n-i, i]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)
A336142
Number of ways to choose a strict composition of each part of a strict integer partition of n.
Original entry on oeis.org
1, 1, 1, 4, 6, 11, 22, 41, 72, 142, 260, 454, 769, 1416, 2472, 4465, 7708, 13314, 23630, 40406, 68196, 119646, 203237, 343242, 586508, 993764, 1677187, 2824072, 4753066, 7934268, 13355658, 22229194, 36945828, 61555136, 102019156, 168474033, 279181966
Offset: 0
The a(1) = 1 through a(5) = 11 ways:
(1) (2) (3) (4) (5)
(1,2) (1,3) (1,4)
(2,1) (3,1) (2,3)
(2),(1) (3),(1) (3,2)
(1,2),(1) (4,1)
(2,1),(1) (3),(2)
(4),(1)
(1,2),(2)
(1,3),(1)
(2,1),(2)
(3,1),(1)
Multiset partitions of partitions are
A001970.
Splittings of partitions are
A323583.
Splittings of partitions with distinct sums are
A336131.
Partitions:
- 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 partition are
A270995.
- Strict compositions of each part of a partition are
A336141.
Strict partitions:
- Partitions of each part of a strict partition are
A271619.
- Compositions of each part of a strict partition are
A304961.
- Strict partitions of each part of a strict partition are
A279785.
- Strict compositions of each part of a strict partition are
A336142.
Compositions:
- Partitions of each part of a composition are
A055887.
- Compositions of each part of a composition are
A133494.
- Strict partitions of each part of a composition are
A304969.
- Strict compositions of each part of a composition are
A307068.
Strict compositions:
- Partitions of each part of a strict composition are
A336342.
- Compositions of each part of a strict composition are
A336127.
- Strict partitions of each part of a strict composition are
A336343.
- Strict compositions of each part of a strict composition are
A336139.
-
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38); # Alois P. Heinz, Jul 31 2020
-
strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[Join@@Permutations/@strptn[#]&/@ctn],{ctn,strptn[n]}]],{n,0,20}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
If[n == 0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0,
If[n == 0, 1, g[n, i-1] + b[i, i, 0]*g[n-i, Min[n-i, i-1]]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)
A358823
Number of odd-length twice-partitions of n into partitions with all odd parts.
Original entry on oeis.org
0, 1, 1, 3, 3, 7, 10, 20, 29, 58, 83, 150, 230, 399, 605, 1037, 1545, 2547, 3879, 6241, 9437, 15085, 22622, 35493, 53438, 82943, 124157, 191267, 284997, 434634, 647437, 979293, 1452182, 2185599, 3228435, 4826596, 7112683, 10575699, 15530404, 22990800, 33651222
Offset: 0
The a(1) = 1 through a(6) = 10 twice-partitions with all odd parts:
(1) (11) (3) (31) (5) (33)
(111) (1111) (311) (51)
(1)(1)(1) (11)(1)(1) (11111) (3111)
(3)(1)(1) (111111)
(11)(11)(1) (3)(11)(1)
(111)(1)(1) (31)(1)(1)
(1)(1)(1)(1)(1) (11)(11)(11)
(111)(11)(1)
(1111)(1)(1)
(11)(1)(1)(1)(1)
The a(1) = 1 through a(6) = 10 twice-partitions into strict partitions:
(1) (2) (3) (4) (5) (6)
(21) (31) (32) (42)
(1)(1)(1) (2)(1)(1) (41) (51)
(2)(2)(1) (321)
(3)(1)(1) (2)(2)(2)
(21)(1)(1) (3)(2)(1)
(1)(1)(1)(1)(1) (4)(1)(1)
(21)(2)(1)
(31)(1)(1)
(2)(1)(1)(1)(1)
This is the odd-length case of
A270995.
Requiring odd sums also gives
A279374 aerated.
This is the case of
A358824 with all odd parts.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
A358334 counts twice-partitions into odd-length partitions.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Flatten[#]]&]],{n,0,10}]
-
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 31 2022
A304786
Expansion of Product_{k>=1} (1 - q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).
Original entry on oeis.org
1, -1, -1, -1, 0, 1, -1, 4, 2, 3, 1, 8, -8, 10, -8, -9, -15, -6, -46, -14, -65, -28, 14, -29, -43, -37, 298, 59, 234, 165, 738, 354, 1083, 703, 1944, -2024, 1917, -1085, 3658, -2385, -6421, -7220, 118, -15569, -11604, -19162, -9448, -36140, -24561, -50505, -24807, 47645
Offset: 0
-
nmax = 51; CoefficientList[Series[Product[(1 - PartitionsQ[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsQ[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 51}]
A327552
Number of partitions in all twice partitions of n where the first partition is strict.
Original entry on oeis.org
0, 1, 2, 7, 11, 29, 63, 125, 225, 489, 930, 1704, 3260, 5859, 10868, 20026, 35062, 61660, 111789, 191119, 337432, 585847, 1003876, 1705380, 2921394, 4930357, 8311554, 14013583, 23435178, 38849655, 64847870, 106784912, 175699558, 289676875, 472418418, 772944773
Offset: 0
a(3) = 7 = 1+1+1+2+2 counting the partitions in 3, 21, 111, 2|1, 11|1.
-
b:= proc(n, i) option remember; `if`(i*(i+1)/2p+[0, p[1]])(
combinat[numbpart](i)*b(n-i, min(n-i, i-1)))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..36);
-
b[n_, i_] := b[n, i] = If[i(i+1)/2Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)
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
A358834
Number of odd-length twice-partitions of n into odd-length partitions.
Original entry on oeis.org
0, 1, 1, 3, 3, 8, 11, 24, 35, 74, 109, 213, 336, 624, 986, 1812, 2832, 5002, 7996, 13783, 21936, 37528, 59313, 99598, 158356, 262547, 415590, 684372, 1079576, 1759984, 2779452, 4491596, 7069572, 11370357, 17841534, 28509802, 44668402, 70975399, 110907748
Offset: 0
The a(1) = 1 through a(6) = 11 twice-partitions:
(1) (2) (3) (4) (5) (6)
(111) (211) (221) (222)
(1)(1)(1) (2)(1)(1) (311) (321)
(11111) (411)
(2)(2)(1) (21111)
(3)(1)(1) (2)(2)(2)
(111)(1)(1) (3)(2)(1)
(1)(1)(1)(1)(1) (4)(1)(1)
(111)(2)(1)
(211)(1)(1)
(2)(1)(1)(1)(1)
The version for set partitions is
A003712.
If the parts are also odd we get
A279374.
The version for multiset partitions of integer partitions is the odd-length case of
A356932, ranked by
A026424 /\
A356935.
This is the odd-length case of
A358334.
This is the odd-lengths case of
A358824.
For odd sums instead of lengths we have
A358826.
The case of odd sums also is the bisection of
A358827.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Length/@#]&]],{n,0,10}]
-
P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022
A358825
Number of ways to choose a sequence of integer partitions, one of each part of an integer partition of n into odd parts.
Original entry on oeis.org
1, 1, 1, 4, 4, 11, 20, 35, 56, 113, 207, 326, 602, 985, 1777, 3124, 5115, 8523, 15011, 24519, 41571, 71096, 115650, 191940, 320651, 530167, 865781, 1442059, 2358158, 3833007, 6325067, 10243259, 16603455, 27151086, 43734197, 71032191, 115091799, 184492464
Offset: 0
The a(1) = 1 through a(5) = 11 twice-partitions:
(1) (1)(1) (3) (3)(1) (5)
(21) (21)(1) (32)
(111) (111)(1) (41)
(1)(1)(1) (1)(1)(1)(1) (221)
(311)
(2111)
(11111)
(3)(1)(1)
(21)(1)(1)
(111)(1)(1)
(1)(1)(1)(1)(1)
For odd parts instead of sums we have
A270995.
For distinct instead of odd sums we have
A271619.
Requiring odd length, odd lengths, and odd parts gives
A279374 aerated.
For odd lengths instead of sums we have
A358334.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Total/@#]&]],{n,0,10}]
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