A358827 -id:A358827

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A358334

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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))

MATHEMATICA

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];

Table[Length[Select[twiptn[n], OddQ[Times@@Length/@#]&]], {n, 0, 10}]

PROG

(PARI)

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

CROSSREFS

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 sums also we have A358827.

For odd length also we have A358834.

A000041 counts integer partitions.

A027193 counts odd-length partitions, ranked by A026424.

A055922 counts partitions with odd multiplicities, also odd parts A117958.

A063834 counts twice-partitions, strict A296122, row-sums of A321449.

Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 01 2022

EXTENSIONS

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

STATUS

approved

A358834

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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)

MATHEMATICA

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];

Table[Length[Select[twiptn[n], OddQ[Length[#]]&&OddQ[Times@@Length/@#]&]], {n, 0, 10}]

PROG

(PARI)

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 04 2022

EXTENSIONS

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

STATUS

approved

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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..28.

Gus Wiseman, Sequences enumerating triangles of integer partitions

EXAMPLE

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)

MATHEMATICA

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];

Table[Length[Select[twiptn[n], OddQ[Length[#]]&&OddQ[Times@@Total/@#]&]], {n, 1, 15, 2}]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 03 2022

STATUS

approved

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