A300301 -id:A300301 - cp's OEIS frontend

A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

1, 1, 2, 6, 12, 30, 82, 198, 533, 1459, 4039, 11634, 34095, 102520, 316456, 995709, 3215552, 10591412, 35633438, 122499429, 428988392, 1532929060, 5579867442, 20677066725, 78027003260, 299413756170, 1168536196157, 4635420192861, 18678567555721, 76451691937279, 317625507668759

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears an odd number of times.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,2,2}}      {{1},{1,2,2}}
                    {{1,2,3}}      {{1},{2,2,2}}
                    {{1},{1},{1}}  {{1},{2,3,3}}
                    {{1},{2},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A001055, A001222, A007716, A298118, A300300, A300301, A318871, A320663, A320665.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A,x,-x))/2)))} \\ Andrew Howroyd, Jan 17 2023

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
J(q, t, k, y)={1/prod(j=1, #q, my(s=q[j], g=gcd(s,t)); (1 + O(x*x^k) - y^(s/g)*x^(s*t/g))^g)}
K(q, t, k) = Vec(J(q,t,k,1)-J(q,t,k,-1), -k)/2
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

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

Gus Wiseman, Dec 04 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			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)

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

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.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A270995, A271619, A279785, A358823, A358837.

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

G.f.: ((1/Product_{k>=1} (1-A027193(k)*x^k)) - (1/Product_{k>=1} (1+A027193(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

A300797 Number of strict trees of weight 2n + 1 in which all outdegrees and all leaves are odd.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 6, 11, 17, 34, 59, 118, 213, 424, 799, 1606, 3072, 6216, 12172, 24650, 48710, 99333, 198237, 405526, 815267, 1673127, 3387165, 6974702, 14179418, 29285048, 59841630, 123848399, 253927322, 526936694, 1084022437, 2253778793, 4649778115

Offset: 0

Gus Wiseman, Mar 13 2018

nonn

A strict tree of weight n > 0 is either a single node of weight n, or a sequence of two or more strict trees with strictly decreasing weights summing to n.

			The a(7) = 6 strict trees: 15, (11 3 1), (9 5 1), (7 5 3), ((7 3 1) 3 1), ((5 3 1) 5 1).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000009, A000992, A032305, A063834, A078408, A089259, A196545, A273873, A279785, A289501, A298118, A300301, A300352, A300353, A300436, A300439, A300440, A300652.

a[n_]:=a[n]=If[OddQ[n],1,0]+Sum[Times@@a/@ptn,{ptn,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&&UnsameQ@@#&]}];
Table[a[n],{n,1,60,2}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))) - prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ Andrew Howroyd, Aug 26 2018

a(30)-a(37) from Alois P. Heinz, Mar 13 2018

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

Gus Wiseman, Dec 03 2022

nonn

			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.
The odd-length case is A358826.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A072233, A279785, A356932, A358824.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Total/@#]&]],{n,0,10}]

G.f.: Product_{k odd} 1/(1-A000041(k)*x^k).

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

Gus Wiseman, Dec 03 2022

nonn

			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)

Gus Wiseman, Sequences enumerating triangles of integer partitions

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.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A072233, A271619, A279785, A356932.

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

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			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)

Gus Wiseman, Sequences enumerating triangles of integer partitions

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.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A072233, A271619, A279785, A306319, A356932, A358824.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&&OddQ[Times@@Total/@#]&]],{n,0,10}]

G.f.: Product_{k odd} 1/(1-A027193(k)*x^k).

A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 107, 336, 1037, 3367, 10924, 36438, 121045, 412789, 1398168, 4831708, 16636297, 58084208, 202101971, 712709423, 2502000811, 8880033929, 31428410158, 112199775788, 399383181020, 1433385148187, 5128572792587, 18481258241133

Offset: 0

Gus Wiseman, Aug 27 2018

nonn

A p-tree of weight n with odd outdegrees is either a single node (if n = 1) or a finite odd-length sequence of at least 3 p-trees with odd outdegrees whose weights are weakly decreasing and sum to n.

			The a(4) = 13 p-trees of weight 9 with odd outdegrees:
  ((((ooo)oo)oo)oo)
  (((ooo)(ooo)o)oo)
  (((ooo)oo)(ooo)o)
  ((ooo)(ooo)(ooo))
  (((ooooo)oo)oo)
  (((ooo)oooo)oo)
  ((ooooo)(ooo)o)
  (((ooo)oo)oooo)
  ((ooo)(ooo)ooo)
  ((ooooooo)oo)
  ((ooooo)oooo)
  ((ooo)oooooo)
  (ooooooooo)

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A027193, A063834, A078408, A196545, A279374, A289501, A298118, A300300, A300301, A300355, A300436, A300647, A300652, A300797, A302243.

b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Table[b[n],{n,1,20,2}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))) - 1/prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ Andrew Howroyd, Aug 27 2018

cp's OEIS Frontend

A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

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A358834 Number of odd-length twice-partitions of n into odd-length partitions.

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A300797 Number of strict trees of weight 2n + 1 in which all outdegrees and all leaves are odd.

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A358825 Number of ways to choose a sequence of integer partitions, one of each part of an integer partition of n into odd parts.

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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.

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A358827 Number of twice-partitions of n into partitions with all odd lengths and sums.

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A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

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