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 */
A358334
Number of twice-partitions of n into odd-length partitions.
Original entry on oeis.org
1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765
Offset: 0
The a(0) = 1 through a(5) = 13 twice-partitions:
() ((1)) ((2)) ((3)) ((4)) ((5))
((1)(1)) ((111)) ((211)) ((221))
((2)(1)) ((2)(2)) ((311))
((1)(1)(1)) ((3)(1)) ((3)(2))
((111)(1)) ((4)(1))
((2)(1)(1)) ((11111))
((1)(1)(1)(1)) ((111)(2))
((211)(1))
((2)(2)(1))
((3)(1)(1))
((111)(1)(1))
((2)(1)(1)(1))
((1)(1)(1)(1)(1))
For multiset partitions of integer partitions:
A356932, ranked by
A356935.
For odd length instead of lengths we have
A358824.
For odd sums instead of lengths we have
A358825.
For odd length also we have
A358834.
A055922 counts partitions with odd multiplicities, also odd parts
A117958.
Cf.
A000219,
A001970,
A072233,
A078408,
A270995,
A279374,
A298118,
A300300,
A300301,
A300647,
A302243.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],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), -(n+1))} \\ Andrew Howroyd, Dec 30 2022
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
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}]
A358826
Number of ways to choose a sequence of partitions, one of each part of an odd-length partition of 2n+1 into odd parts.
Original entry on oeis.org
1, 4, 11, 35, 113, 326, 985, 3124, 8523, 24519, 71096, 191940, 530167, 1442059, 3833007, 10243259, 27151086, 71032191, 184492464, 478339983, 1227208513, 3140958369, 8016016201, 20210235189, 50962894061, 127936646350, 319022819270, 794501931062, 1969154638217
Offset: 0
The a(1) = 1 through a(5) = 11 twice-partitions:
(1) (3) (5)
(21) (32)
(111) (41)
(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 length and sums we have
A270995.
Requiring odd lengths and odd parts gives
A279374 aerated.
This is the case of
A358824 with odd sums.
This is the odd-length case (hence odd bisection) of
A358825.
For odd lengths (instead of length) we have
A358827.
For odd lengths instead of sums we have
A358834.
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@@Total/@#]&]],{n,1,15,2}]
A358827
Number of twice-partitions of n into partitions with all odd lengths and sums.
Original entry on oeis.org
1, 1, 1, 3, 3, 7, 11, 19, 27, 51, 83, 128, 208, 324, 542, 856, 1332, 2047, 3371, 5083, 8009, 12545, 19478, 29770, 46038, 70777, 108627, 167847, 255408, 388751, 593475, 901108, 1361840, 2077973, 3125004, 4729056, 7146843, 10732799, 16104511, 24257261, 36305878
Offset: 0
The a(1) = 1 through a(6) = 11 twice-partitions:
(1) (1)(1) (3) (3)(1) (5) (3)(3)
(111) (111)(1) (221) (5)(1)
(1)(1)(1) (1)(1)(1)(1) (311) (111)(3)
(11111) (221)(1)
(3)(1)(1) (3)(111)
(111)(1)(1) (311)(1)
(1)(1)(1)(1)(1) (111)(111)
(11111)(1)
(3)(1)(1)(1)
(111)(1)(1)(1)
(1)(1)(1)(1)(1)(1)
This is the case of
A358334 with odd sums.
This is the case of
A358825 with odd lengths.
The case of odd length is the odd bisection.
For odd parts instead of lengths and sums we have
A270995.
Requiring odd parts also gives
A279374 aerated.
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@@Length/@#]&&OddQ[Times@@Total/@#]&]],{n,0,10}]
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.
-
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}]
-
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
A358828
Number of twice-partitions of n with no singletons.
Original entry on oeis.org
1, 0, 1, 2, 5, 8, 19, 30, 68, 111, 229, 380, 799, 1280, 2519, 4325, 8128, 13666, 25758, 43085, 79300, 134571, 240124, 407794, 730398, 1224821, 2152122, 3646566, 6338691, 10657427, 18469865, 30913539, 53108364, 88953395, 151396452, 253098400, 429416589
Offset: 0
The a(2) = 1 through a(6) = 19 twice-partitions:
(11) (21) (22) (32) (33)
(111) (31) (41) (42)
(211) (221) (51)
(1111) (311) (222)
(11)(11) (2111) (321)
(11111) (411)
(21)(11) (2211)
(111)(11) (3111)
(21111)
(111111)
(21)(21)
(22)(11)
(31)(11)
(111)(21)
(21)(111)
(211)(11)
(111)(111)
(1111)(11)
(11)(11)(11)
The version for multiset partitions of integer partitions is
A304966.
Allowing singletons other than (1) gives
A358829.
A002865 counts partitions with no 1's.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],FreeQ[Length/@#,1]&]],{n,0,10}]
A358829
Number of twice-partitions of n with no (1)'s.
Original entry on oeis.org
1, 0, 2, 3, 9, 13, 38, 56, 144, 237, 524, 886, 1961, 3225, 6700, 11702, 23007, 39787, 77647, 133707, 254896, 442736, 820703, 1427446, 2630008, 4535330, 8224819, 14250148, 25513615, 43981753, 78252954, 134323368, 236900355, 406174046, 709886932, 1213934012
Offset: 0
The a(2) = 2 through a(5) = 13 twice-partitions:
(2) (3) (4) (5)
(11) (21) (22) (32)
(111) (31) (41)
(211) (221)
(1111) (311)
(2)(2) (2111)
(11)(2) (3)(2)
(2)(11) (11111)
(11)(11) (21)(2)
(3)(11)
(111)(2)
(21)(11)
(111)(11)
The version for multiset partitions of integer partitions is
A317911.
Forbidding all singletons gives
A358828.
A002865 counts partitions with no 1's.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],FreeQ[Total/@#,1]&]],{n,0,10}]
Showing 1-10 of 10 results.
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