A303546 -id:A303546 - cp's OEIS frontend

A303551 Number of aperiodic multisets of compositions of total weight n.

1, 2, 6, 15, 41, 95, 243, 567, 1366, 3189, 7532, 17428, 40590, 93465, 215331, 493150, 1127978, 2569049, 5841442, 13240351, 29953601, 67596500, 152258270, 342235866, 767895382, 1719813753, 3845442485, 8584197657, 19133459138, 42583565928, 94641591888

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

			The a(4) = 15 aperiodic multisets of compositions are:
{4}, {31}, {22}, {211}, {13}, {121}, {112}, {1111},
{1,3}, {1,21}, {1,12}, {1,111}, {2,11},
{1,1,2}, {1,1,11}.
Missing from this list are {1,1,1,1}, {2,2}, and {11,11}.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000740, A000837, A007716, A007916, A034691, A100953, A255906, A269134, A301700, A303386, A303431, A303546, A303552.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*2^(d-1), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(mobius(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..35);  # Alois P. Heinz, Apr 26 2018

nn=20;
ser=Product[1/(1-x^n)^2^(n-1),{n,nn}]
Table[Sum[MoebiusMu[d]*SeriesCoefficient[ser,{x,0,n/d}],{d,Divisors[n]}],{n,1,nn}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=EulerT(vector(n, n, 2^(n-1)))); vector(n, n, sumdiv(n, d, moebius(d)*u[n/d]))} \\ Andrew Howroyd, Sep 15 2018

a(n) = Sum_{d|n} mu(d) * A034691(n/d).

A320806 Number of non-isomorphic multiset partitions of weight n in which each of the parts and each part of the dual, as well as both the multiset union of the parts and the multiset union of the dual parts, is an aperiodic multiset.

Original entry on oeis.org

1, 1, 1, 4, 10, 39, 81, 343, 903, 3223, 9989

Offset: 0

Gus Wiseman, Nov 07 2018

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the positive entries in each row and column are relatively prime and (2) the row sums and column sums are relatively prime.
The last condition (aperiodicity of the multiset union of the dual) is equivalent to the parts having relatively prime sizes.
A multiset is aperiodic if its multiplicities are relatively prime.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 10 multiset partitions:
  {{1}}  {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}
                    {{2},{1,2}}    {{2},{1,2,2}}
                    {{1},{2},{2}}  {{3},{1,2,3}}
                    {{1},{2},{3}}  {{1},{1},{2,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810, A321283.

A320811 Number of non-isomorphic multiset partitions with no singletons of aperiodic multisets of size n.

Original entry on oeis.org

1, 0, 1, 2, 7, 21, 57, 200, 575, 1898, 5893

Offset: 0

Gus Wiseman, Nov 08 2018

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the row sums are all > 1 and (2) the column sums are relatively prime.
A multiset is aperiodic if its multiplicities are relatively prime.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 21 multiset partitions:
  {{1,2}}  {{1,2,2}}  {{1,2,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,3,3}}    {{1,2,2,2,2}}
                      {{1,2,3,4}}    {{1,2,2,3,3}}
                      {{1,2},{2,2}}  {{1,2,3,3,3}}
                      {{1,2},{3,3}}  {{1,2,3,4,4}}
                      {{1,2},{3,4}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,1},{1,2,2}}
                                     {{1,1},{2,2,2}}
                                     {{1,1},{2,3,3}}
                                     {{1,1},{2,3,4}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{2,3},{1,2,3}}
                                     {{3,3},{1,2,3}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813, A321283, A321390.

A320812 Number of non-isomorphic aperiodic multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 10, 23, 79, 204, 670, 1974, 6521, 21003, 71944, 248055, 888565, 3240552, 12152093, 46527471, 182337383, 729405164, 2979114723, 12407307929, 52670334237, 227725915268, 1002285201807, 4487915293675, 20434064047098, 94559526594316, 444527729321513

Offset: 0

Gus Wiseman, Nov 08 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 2 through a(5) = 23 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{2,2}}  {{1,2,3,4,4}}
                      {{1,2},{2,2}}  {{1,2,3,4,5}}
                      {{1,2},{3,3}}  {{1,1},{1,1,1}}
                      {{1,2},{3,4}}  {{1,1},{1,2,2}}
                      {{1,3},{2,3}}  {{1,1},{2,2,2}}
                                     {{1,1},{2,3,3}}
                                     {{1,1},{2,3,4}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{2,3},{1,2,3}}
                                     {{3,3},{1,2,3}}

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

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813, A321390.

a(n) = Sum_{d|n} mu(d)*A302545(n/d) for n > 0. - Andrew Howroyd, Jan 16 2023

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

A303974 Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.

Original entry on oeis.org

1, 2, 1, 3, 3, 3, 4, 6, 10, 6, 5, 10, 22, 23, 15, 6, 15, 40, 57, 62, 27, 7, 21, 65, 115, 165, 129, 63, 8, 28, 98, 205, 356, 385, 318, 120, 9, 36, 140, 336, 676, 914, 1005, 676, 252, 10, 45, 192, 518, 1176, 1885, 2524, 2334, 1524, 495, 11, 55, 255, 762, 1918, 3528, 5495, 6319, 5607, 3261, 1023

Offset: 1

Gus Wiseman, May 03 2018

A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.

			Triangle begins:
1
2    1
3    3    3
4    6   10    6
5   10   22   23   15
6   15   40   57   62   27
7   21   65  115  165  129   63
8   28   98  205  356  385  318  120
9   36  140  336  676  914 1005  676  252
The a(4,3) = 10 multisets: (112), (113), (122), (123), (124), (133), (134), (223), (233), (234).
The a(5,4) = 23 multisets:
(1112), (1222),
(1113), (1123), (1223), (1233), (1333), (2223), (2333),
(1124), (1134), (1224), (1234), (1244), (1334), (1344), (2234), (2334), (2344),
(1235), (1245), (1345), (2345).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A303976.

Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Length],{n,10}]

T(n,k)={sumdiv(k, d, moebius(k/d)*sum(i=1, d, binomial(d-1, i-1)*binomial(n-k+i, i)))} \\ Andrew Howroyd, Sep 18 2018

T(n,k) = Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018

Terms a(56) and beyond from Andrew Howroyd, Sep 18 2018

A320802 Number of non-isomorphic aperiodic multiset partitions of weight n whose dual is also an aperiodic multiset partition.

Original entry on oeis.org

1, 1, 2, 8, 26, 89, 274, 908, 2955, 9926, 34021, 119367, 428612, 1574222, 5914324, 22699632, 88997058, 356058538, 1453059643, 6044132792, 25612530061, 110503625785, 485161109305, 2166488899640, 9835209048655, 45370059225137, 212582814591083, 1011306624492831

Offset: 0

Gus Wiseman, Nov 06 2018

nonn

Also the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns where the multiset of rows and the multiset of columns are both aperiodic, up to row and column permutations.
A multiset is aperiodic if its multiplicities are relatively prime.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
Also the number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods, where the period of a multiset is the GCD of its multiplicities.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 26 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1,2,2}}      {{1,2,2,2}}
                    {{1},{1,1}}    {{1,2,3,3}}
                    {{1},{2,2}}    {{1},{1,1,1}}
                    {{1},{2,3}}    {{1},{1,2,2}}
                    {{2},{1,2}}    {{1,1},{2,2}}
                    {{1},{2},{2}}  {{1},{2,2,2}}
                    {{1},{2},{3}}  {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Jinyuan Wang, Table of n, a(n) for n = 0..50

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320800-A320810.

Second Moebius transform of A007716, or Moebius transform of A303546, where the Moebius transform of a sequence b is a(n) = Sum_{d|n} mu(d) * b(n/d).

a(26)-a(27) from Jinyuan Wang, Jun 27 2020

A320803 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic multisets.

Original entry on oeis.org

1, 1, 3, 7, 21, 56, 174, 517, 1664, 5383, 18199, 62745, 223390, 813425, 3040181, 11620969, 45446484, 181537904, 740369798, 3079779662, 13059203150, 56406416004, 248027678362, 1109626606188, 5048119061134, 23342088591797, 109648937760252, 523036690273237

Offset: 0

Gus Wiseman, Nov 06 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions with aperiodic parts:
  {{1}}  {{1,2}}    {{1,2,2}}      {{1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,3,3}}
         {{1},{2}}  {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{1,2,2}}
                    {{1},{1},{1}}  {{1,2},{1,2}}
                    {{1},{2},{2}}  {{1},{2,3,3}}
                    {{1},{2},{3}}  {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{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. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A303709, A303710, A320800-A320810.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(mbt=vector(n, d, moebius(d)), s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(dirmul(mbt, sum(t=1, n, K(q, t, n)/t)))), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2023

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

A320809 Number of non-isomorphic multiset partitions of weight n in which each part and each part of the dual, as well as the multiset union of the parts, is an aperiodic multiset.

Original entry on oeis.org

1, 1, 2, 5, 13, 40, 99, 344, 985, 3302, 10583

Offset: 0

Gus Wiseman, Nov 07 2018

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the positive entries in each row and column are relatively prime and (2) the column sums are relatively prime.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset is aperiodic if its multiplicities are relatively prime.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 13 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}
                    {{2},{1,2}}    {{1,2},{3,4}}
                    {{1},{2},{2}}  {{1,3},{2,3}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810, A321283.

A321390 Third Moebius transform of A007716. Number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods and whose dual is also an aperiodic multiset partition.

Original entry on oeis.org

1, 1, 1, 7, 24, 88, 265, 907, 2929, 9918, 33931, 119366, 428314, 1574221, 5913415, 22699536, 88994103, 356058537, 1453049451, 6044132791, 25612496016, 110503624870, 485160989937, 2166488899639, 9835208617114, 45370059225048

Offset: 0

Gus Wiseman, Nov 08 2018

nonn

The Moebius transform c of a sequence b is c(n) = Sum_{d|n} mu(d) * b(n/d).
Also the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns where the multiset of rows and the multiset of columns are both aperiodic and the nonzero entries are relatively prime, up to row and column permutations.
A multiset is aperiodic if its multiplicities are relatively prime. The period of a multiset is the GCD of its multiplicities.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 24 multiset partitions:
  {{1}}  {{1},{2}}  {{1,2,2}}      {{1,2,2,2}}
                    {{1},{1,1}}    {{1,2,3,3}}
                    {{1},{2,2}}    {{1},{1,1,1}}
                    {{1},{2,3}}    {{1},{1,2,2}}
                    {{2},{1,2}}    {{1},{2,2,2}}
                    {{1},{2},{2}}  {{1,2},{2,2}}
                    {{1},{2},{3}}  {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320800-A320810.

A303976 Number of different aperiodic multisets that fit within some normal multiset of size n.

Original entry on oeis.org

1, 3, 9, 26, 75, 207, 565, 1518, 4044, 10703, 28234, 74277, 195103, 511902, 1342147, 3517239, 9214412, 24134528, 63204417, 165505811, 433361425, 1134664831, 2970787794, 7777975396, 20363634815, 53313819160, 139579420528, 365427311171, 956707667616, 2504704955181

Offset: 1

Gus Wiseman, May 03 2018

nonn

A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.

			The a(4) = 26 aperiodic multisets:
(1), (2), (3), (4),
(12), (13), (14), (23), (24), (34),
(112), (113), (122), (123), (124), (133), (134), (223), (233), (234),
(1112), (1123), (1222), (1223), (1233), (1234).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Row sums of A303974.

Cf. A000740, A000837, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&]],{n,10}]

seq(n)={Vec(sum(d=1, n, moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x))} \\ Andrew Howroyd, Feb 04 2021

a(n) = Sum_{k=1..n} Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018

G.f.: Sum_{d>=1} mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)). - Andrew Howroyd, Feb 04 2021

Terms a(13) and beyond from Andrew Howroyd, Sep 18 2018

cp's OEIS Frontend

A303551 Number of aperiodic multisets of compositions of total weight n.

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A320806 Number of non-isomorphic multiset partitions of weight n in which each of the parts and each part of the dual, as well as both the multiset union of the parts and the multiset union of the dual parts, is an aperiodic multiset.

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A320811 Number of non-isomorphic multiset partitions with no singletons of aperiodic multisets of size n.

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A320812 Number of non-isomorphic aperiodic multiset partitions of weight n with no singletons.

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A303974 Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.

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A320802 Number of non-isomorphic aperiodic multiset partitions of weight n whose dual is also an aperiodic multiset partition.

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A320803 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic multisets.

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A320809 Number of non-isomorphic multiset partitions of weight n in which each part and each part of the dual, as well as the multiset union of the parts, is an aperiodic multiset.

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A321390 Third Moebius transform of A007716. Number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods and whose dual is also an aperiodic multiset partition.