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a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d.
+10
57
1, 3, 4, 8, 8, 17, 16, 30, 34, 52, 57, 99, 102, 153, 187, 261, 298, 432, 491, 684, 811, 1061, 1256, 1696, 1966, 2540, 3044, 3876, 4566, 5846, 6843, 8610, 10203, 12610, 14906, 18491, 21638, 26508, 31290, 38044, 44584, 54133, 63262, 76241
COMMENTS
Inverse Moebius transform of A000041. Sum of the partition numbers of the divisors of n. - Omar E. Pol, Feb 25 2014 Number of constant multiset partitions of multisets spanning an initial interval of positive integers with multiplicities an integer partition of n. - Gus Wiseman, Sep 16 2018
FORMULA
G.f.: Sum_{k>0} (-1+1/Product_{i>0} (1-z^(k*i))). - Vladeta Jovovic, Jun 22 2003
EXAMPLE
For n = 10 the divisors of 10 are 1, 2, 5, 10, hence the partition numbers of the divisors of 10 are 1, 2, 7, 42, so a(10) = 1 + 2 + 7 + 42 = 52. - Omar E. Pol, Feb 26 2014 The a(6) = 17 constant multiset partitions:
(111111) (111)(111) (11)(11)(11) (1)(1)(1)(1)(1)(1)
(111222) (12)(12)(12)
(111122) (112)(112)
(112233) (123)(123)
(111112)
(111123)
(111223)
(111234)
(112234)
(112345)
(123456)
(End)
MAPLE
with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l) do c := c+numbpart(l[i]) od: RETURN(c): end: for j from 1 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007
MATHEMATICA
a[n_] := Sum[ PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Oct 03 2013 *)
Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.
+10
10
0, 2, 3, 12, 7, 84, 15, 410, 354, 3073, 56, 28300, 101, 210036, 126839, 2070047, 297, 25295952, 490, 269662769, 89071291, 3449056162, 1255, 51132696310, 400625539, 713071048480, 145126661415, 11351097702297, 4565, 199926713003444, 6842, 3460838122540969
COMMENTS
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 the column sums are not relatively prime.
Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is periodic, where a multiset is periodic if its multiplicities have a common divisor > 1.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
EXAMPLE
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions whose part-sizes have a common divisor:
{{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},{1,1}} {{1,2,3,4,4}}
{{1,1},{2,2}} {{1,2,3,4,5}}
{{1,2},{1,2}}
{{1,2},{2,2}}
{{1,2},{3,3}}
{{1,2},{3,4}}
{{1,3},{2,3}}
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions with periodic multiset union:
{{1,1}} {{1,1,1}} {{1,1,1,1}} {{1,1,1,1,1}}
{{1},{1}} {{1},{1,1}} {{1,1,2,2}} {{1},{1,1,1,1}}
{{1},{1},{1}} {{1},{1,1,1}} {{1,1},{1,1,1}}
{{1,1},{1,1}} {{1},{1},{1,1,1}}
{{1},{1,2,2}} {{1},{1,1},{1,1}}
{{1,1},{2,2}} {{1},{1},{1},{1,1}}
{{1,2},{1,2}} {{1},{1},{1},{1},{1}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{2},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
PROG
(PARI) \\ See links in A339645 for combinatorial species functions. seq(n)={my(A=symGroupSeries(n)); Vec(OgfSeries(sCartProd(sExp(A), -sum(d=2, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A, i*d)*x^(i*d)))) ))), -n)} \\ Andrew Howroyd, Jan 17 2023
CROSSREFS
Cf. A007716, A018783, A047966, A120733, A303546, A303547, A305563, A319149, A319162, A319164, A319810.
Number of integer partitions of n that are neither relatively prime nor aperiodic.
+10
8
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
COMMENTS
A partition is aperiodic if its multiplicities are relatively prime.
EXAMPLE
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).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And[GCD@@#>1, GCD@@Length/@Split[#]>1]&]], {n, 30}]
Perfect powers whose prime multiplicities appear with relatively prime multiplicities.
+10
7
4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 144, 169, 243, 256, 289, 324, 343, 361, 400, 512, 529, 576, 625, 729, 784, 841, 961, 1024, 1331, 1369, 1600, 1681, 1728, 1849, 1936, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2704, 2809, 2916, 3125
COMMENTS
Perfect powers n such that A181819(n) is not a perfect power (i.e. belongs to A007916).
EXAMPLE
The sequence of integer partitions whose Heinz numbers are in the sequence begins: (11), (111), (22), (1111), (33), (222), (11111), (44), (111111), (2222), (55), (333), (1111111), (221111), (66), (22222), (11111111), (77), (222211), (444), (88), (331111), (111111111), (99), (22111111), (3333), (222222), (441111).
MATHEMATICA
Select[Range[1000], And[GCD@@FactorInteger[#][[All, 2]]>1, GCD@@Length/@Split[Sort[FactorInteger[#][[All, 2]]]]==1]&]
CROSSREFS
Cf. A001597, A007916, A056239, A072774, A181819, A289509, A296150, A298748, A319151, A319161, A319162, A319165.
Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.
+10
6
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
COMMENTS
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)
EXAMPLE
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).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]], {n, 30}]
CROSSREFS
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.
Number of fully periodic integer partitions of n.
+10
5
1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 2, 8, 7, 11, 2, 17, 2, 18, 9, 15, 2, 32, 5, 22, 12, 34, 2, 54, 2, 49, 16, 51, 10, 94, 2, 77, 23, 112, 2, 152, 2, 148, 47, 165, 2, 258, 7, 247, 52, 286, 2, 400, 17, 402, 78, 439, 2, 657, 2, 594, 131, 711, 24
COMMENTS
An integer partition is fully periodic iff either it is a singleton or it is a periodic partition (meaning its multiplicities have a common divisor > 1) with fully periodic multiplicities.
EXAMPLE
The a(12) = 11 fully periodic integer partitions:
(12)
(6,6)
(4,4,4)
(5,5,1,1)
(4,4,2,2)
(3,3,3,3)
(3,3,3,1,1,1)
(3,3,2,2,1,1)
(2,2,2,2,2,2)
(2,2,2,2,1,1,1,1)
(1,1,1,1,1,1,1,1,1,1,1,1)
Periodic partitions missing from this list are:
(4,4,1,1,1,1)
(3,3,1,1,1,1,1,1)
(2,2,2,1,1,1,1,1,1)
(2,2,1,1,1,1,1,1,1,1)
The first non-uniform fully periodic partition is (4,4,3,3,2,2,2,2,1,1,1,1).
The first periodic integer partition that is not fully periodic is (2,2,1,1,1,1).
MATHEMATICA
totperQ[m_]:=Or[Length[m]==1, And[GCD@@Length/@Split[Sort[m]]>1, totperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n], totperQ]], {n, 30}]
CROSSREFS
Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319811.
Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.
+10
4
0, 0, 0, 1, 0, 3, 0, 4, 2, 7, 0, 13, 0, 15, 8, 21, 0, 37, 0, 44, 16, 56, 0, 93, 6, 101, 29, 137, 0, 217, 0, 230, 57, 297, 20, 450, 0, 490, 102, 643, 0, 918, 0, 1004, 202, 1255, 0, 1783, 14, 1992, 298, 2438, 0, 3364, 61, 3734, 491, 4565, 0, 6251, 0, 6842, 818
COMMENTS
The singleton and the all-ones partitions are ignored, so that a(n)=0 if n is prime. If a partition is listed as m_1^am_2^bm_3^c..., then it is counted exactly when gcd(a,b,c,...)>1. These are equinumerous (conjugate) with those partitions for which gcd(m_1,m_2,...)>1 (less 1, the singleton), hence the formula.
EXAMPLE
a(25) = 6: 1^(15)2^5 = 5{1, 1, 1, 2}, 1^52^(10) = 5{1, 2, 2}, 1^(10)3^5 = 5{3, 1, 1}, 2^53^5 = 5{3, 2}, 1^44^4 = 5{4, 1}, 5^5 = 5{5}.
Note that A000041(25)=P(25)=1958, only 6 of which satisfy the criterion.
MAPLE
with(combinat):PartMulti:=proc(n::nonnegint) local count, a, i, j, b, m, k, part_vec;
bigcount:=0; if isprime(n) then return(bigcount) else ps:=partition(n); b:=nops(ps);
for m from 2 to b-1 do p:=ps[m]; a:=nops(p); part_vec:=array(1..n);
for k from 1 to n do part_vec[k]:=0 od;
for i from 1 to a do j:=p[i]; part_vec[j]:=part_vec[j]+1 od;
g:=0; for j from 1 to n do g:=igcd(g, part_vec[j]) od;
if g>1 then bigcount:=bigcount+1 fi od; return(bigcount) end if end proc;
seq(PartMulti(q), q=1..49);
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And[Length[#]<n, GCD@@Length/@Split[#]>1]&]], {n, 20}] (* Gus Wiseman, Dec 06 2018 *)
CROSSREFS
Cf. A000837, A018783, A047966, A100953, A303386, A303547, A303553, A319149, A319162, A319164, A319810.
Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime.
+10
2
0, 1, 1, 1, 1, 2, 1, 3, 2, 6, 1, 9, 1, 14, 7, 17, 1, 32, 1, 36, 15, 55, 1, 77, 6, 100, 27, 121, 1, 200, 1, 209, 56, 296, 19, 403, 1, 489, 101, 596, 1, 885, 1, 947, 192, 1254, 1, 1673, 14, 1979, 297, 2336, 1, 3300, 60, 3594, 490, 4564, 1, 5988, 1, 6841, 800
COMMENTS
An integer partition is aperiodic if its multiplicities are relatively prime.
EXAMPLE
The a(12) = 9 integer partitions that are relatively prime but not aperiodic:
(5511),
(332211), (333111), (441111),
(22221111), (33111111),
(222111111),
(2211111111),
(111111111111).
The a(12) = 9 integer partitions that are aperiodic but not relatively prime:
(12),
(8,4), (9,3), (10,2),
(6,3,3), (6,4,2), (8,2,2),
(6,2,2,2),
(4,2,2,2,2).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And[GCD@@#==1, GCD@@Length/@Split[#]>1]&]], {n, 30}]
Number of totally aperiodic integer partitions of n.
+10
1
1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 99, 117, 162, 203, 286, 333, 469, 558, 737, 903, 1196, 1414, 1860, 2232, 2839, 3422, 4359, 5144, 6531, 7762, 9617, 11479, 14182, 16715, 20630, 24333, 29569, 34890, 42335, 49515, 59871, 70042, 83810, 98105, 117152
COMMENTS
An integer partition is totally aperiodic iff either it is strict or it is aperiodic with totally aperiodic multiplicities.
EXAMPLE
The a(6) = 7 aperiodic integer partitions are: (6), (51), (42), (411), (321), (3111), (21111). The first aperiodic integer partition that is not totally aperiodic is (432211).
MATHEMATICA
totaperQ[m_]:=Or[UnsameQ@@m, And[GCD@@Length/@Split[Sort[m]]==1, totaperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n], totaperQ]], {n, 30}]
CROSSREFS
Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319810.
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