A100953 -id:A100953 - cp's OEIS frontend

A319162 Number of periodic integer partitions of n whose multiplicities are aperiodic, meaning the multiplicities of these multiplicities are relatively prime.

0, 1, 1, 2, 1, 3, 1, 4, 2, 6, 1, 9, 1, 12, 6, 16, 1, 27, 1, 33, 12, 46, 1, 70, 5, 84, 22, 110, 1, 172, 1, 188, 46, 251, 15, 366, 1, 418, 84, 540, 1, 775, 1, 863, 162, 1095, 1, 1535, 11, 1750, 251, 2154, 1, 2963, 49, 3323, 418, 4106, 1, 5567

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

			The a(12) = 9 partitions:
  (66),
  (444), (441111),
  (3333), (33111111),
  (222222), (222111111), (2211111111),
  (111111111111).

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319163, A319164.

Table[Length[Select[IntegerPartitions[n],And[GCD@@Sort[Length/@Split[#]]>1,GCD@@Length/@Split[Sort[Length/@Split[#]]]==1]&]],{n,30}]

A320800 Number of non-isomorphic multiset partitions of weight n in which both the multiset union of the parts and the multiset union of the dual parts are aperiodic.

Original entry on oeis.org

1, 1, 1, 5, 14, 78, 157, 881, 2267, 9257, 28397

Offset: 0

Gus Wiseman, Nov 02 2018

The latter condition 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) = 14 multiset partitions:
  {{1}}  {{1},{2}}  {{1},{2,2}}    {{1},{2,2,2}}
                    {{1},{2,3}}    {{1},{2,3,3}}
                    {{2},{1,2}}    {{1},{2,3,4}}
                    {{1},{2},{2}}  {{2},{1,2,2}}
                    {{1},{2},{3}}  {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{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, A320801-A320810.

A319164 Number of integer partitions of n that are neither relatively prime nor aperiodic.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 5, 0, 2, 2, 5, 0, 6, 0, 9, 2, 2, 0, 17, 1, 2, 3, 17, 0, 18, 0, 22, 2, 2, 2, 48, 0, 2, 2, 48, 0, 34, 0, 58, 11, 2, 0, 111, 1, 14, 2, 103, 0, 65, 2, 141, 2, 2, 0, 264, 0, 2, 19, 231, 2, 116, 0, 299, 2, 42

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

A partition is aperiodic if its multiplicities are relatively prime.

			The a(24) = 17 integer partitions:
  (12,12),
  (8,8,8),
  (6,6,6,6), (8,8,4,4), (9,9,3,3), (10,10,2,2),
  (4,4,4,4,4,4), (6,6,3,3,3,3), (6,6,4,4,2,2), (6,6,6,2,2,2), (8,8,2,2,2,2),
  (3,3,3,3,3,3,3,3), (4,4,4,4,2,2,2,2), (6,6,2,2,2,2,2,2),
  (4,4,4,2,2,2,2,2,2),
  (4,4,2,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2,2,2,2).

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319165.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#>1,GCD@@Length/@Split[#]>1]&]],{n,30}]

A318745 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being coprime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 19, 32, 53, 94, 158, 279, 480, 847, 1487, 2647, 4676, 8349, 14865, 26630, 47700, 85778, 154290, 278318, 502437, 908880, 1645713, 2984546, 5417743, 9847189, 17914494, 32625523, 59467893, 108493134, 198089610, 361965238, 661883231, 1211161991

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(7) = 12 Lyndon compositions with adjacent parts coprime:
  (7)
  (16) (25) (34)
  (115)
  (1114) (1213) (1132) (1123)
  (11113) (11212)
  (111112)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A100953, A167606, A296302, A318728, A318731, A318746, A318747.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,LyndonQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->gcd(i, j)==1))); vector(n, n, (n > 1) + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

a(n) = A328669(n) + 1 for n > 1. - Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A320804 Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.

Original entry on oeis.org

1, 0, 1, 2, 6, 13, 41, 104, 326, 958, 3096, 9958, 33869, 116806, 417741, 1526499, 5732931, 22015642, 86543717, 347495480, 1424832602, 5959123908, 25407212843, 110344848622, 487879651220, 2194697288628, 10039367091586, 46675057440634, 220447539120814

Offset: 0

Gus Wiseman, Nov 06 2018

nonn

Also the number of nonnegative integer matrices with (1) sum of elements equal to n, (2) no zero columns, (3) no rows summing to 0 or 1, and (4) no rows whose nonzero entries have a common divisor > 1, up to row and column permutations.
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) = 13 multiset partitions with aperiodic parts and no singletons:
  {{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},{1,2}}  {{1,2,3,3,3}}
                      {{1,2},{3,4}}  {{1,2,3,4,4}}
                      {{1,3},{2,3}}  {{1,2,3,4,5}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{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.

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))}
S(q, t, k)={Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t))  + 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)) - sum(t=1, n, S(q, t, n)/t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2023

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

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

Original entry on oeis.org

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).

A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 45, 58, 83, 108, 142, 188, 250, 315, 417, 528, 674, 861, 1094, 1363, 1724, 2152, 2670, 3311, 4105, 5021, 6193, 7561, 9216, 11219, 13614, 16419, 19886, 23920, 28733, 34438, 41272, 49184, 58746, 69823, 82948, 98380, 116567

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

From Gus Wiseman, Jul 11 2023: (Start)
A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.
For example:
- The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (3,2,2,1) are (2,1,1), with multiplicities (2,1), which are relatively prime, so it is counted under a(8).
- The multiplicities of (3,3,1,1) are (2,2), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (4,4,4,3,3,3,2,1) are (3,3,1,1), with multiplicities (2,2), which have common divisor 2, so it is not counted under a(24).
(End)

			The a(8) = 16 partitions:
  (8),
  (44),
  (332), (422), (611),
  (2222), (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111),
  (11111111).
Missing from this list are: (53), (62), (71), (431), (521), (3311).

These partitions have ranks A319161.
For distinct instead of relatively prime multiplicities we have A325329.
Cf. A000837, A001597, A007916, A047966, A071625, A098859, A100953, A181819, A182850, A182857, A305563, A319149, A319162, A319164.

Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]],{n,30}]

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

cp's OEIS Frontend

A319162 Number of periodic integer partitions of n whose multiplicities are aperiodic, meaning the multiplicities of these multiplicities are relatively prime.

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A320800 Number of non-isomorphic multiset partitions of weight n in which both the multiset union of the parts and the multiset union of the dual parts are aperiodic.

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A319164 Number of integer partitions of n that are neither relatively prime nor aperiodic.

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A318745 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being coprime.

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A320804 Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.

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

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A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

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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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