A319810 -id:A319810 - cp's OEIS frontend

A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

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

Offset: 1

Gus Wiseman, Nov 15 2018

nonn

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.

			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}}

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A007716, A018783, A047966, A120733, A303546, A303547, A305563, A319149, A319162, A319164, A319810.

\\ 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

a(n) = A007716(n) - A321283(n). - Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A108572 Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.

Original entry on oeis.org

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

Offset: 1

Len Smiley, Jul 25 2005

nonn

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.

			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.

Cf. A000837, A018783, A047966, A100953, A303386, A303547, A303553, A319149, A319162, A319164, A319810.

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);

Table[Length[Select[IntegerPartitions[n],And[Length[#]1]&]],{n,20}] (* Gus Wiseman, Dec 06 2018 *)

a(n) = A018783(n)-1, n>1. - Vladeta Jovovic, Jul 28 2005

More terms from Gus Wiseman, Dec 06 2018

A325332 Number of totally abnormal integer partitions of n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 8, 1, 7, 5, 10, 2, 16, 4, 21, 15, 24, 17, 49, 29, 53, 53, 84, 65, 121, 92, 148, 141, 186, 179, 280, 223, 317, 318, 428, 387, 576, 512, 700, 734, 899, 900, 1260, 1207, 1551, 1668, 2041, 2109, 2748, 2795, 3463, 3775, 4446

Offset: 0

Gus Wiseman, May 01 2019

nonn

A multiset is normal if its union is an initial interval of positive integers. A multiset is totally abnormal if it is not normal and either it is a singleton or its multiplicities form a totally abnormal multiset.

The Heinz numbers of these partitions are given by A325372.

			The a(2) = 1 through a(12) = 8 totally abnormal partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)   (C)
            (22)       (33)        (44)    (333)  (55)           (66)
                       (222)       (2222)         (3322)         (444)
                                   (3311)         (4411)         (3333)
                                                  (22222)        (4422)
                                                                 (5511)
                                                                 (222222)
                                                                 (333111)

Cf. A181819, A275870, A305563, A317088, A317245, A317491, A317589, A319149, A319810, A325372.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
antinrmQ[ptn_]:=!normQ[ptn]&&(Length[ptn]==1||antinrmQ[Sort[Length/@Split[ptn]]]);
Table[Length[Select[IntegerPartitions[n],antinrmQ]],{n,0,30}]

A319811 Number of totally aperiodic integer partitions of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 28 2018

nonn

An integer partition is totally aperiodic iff either it is strict or it is aperiodic with totally aperiodic multiplicities.

			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).

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319810.

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}]

A325373 Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.

Original entry on oeis.org

9, 25, 27, 49, 81, 100, 121, 125, 169, 196, 225, 243, 289, 343, 361, 441, 484, 529, 625, 676, 729, 841, 961, 1000, 1089, 1156, 1225, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2187, 2197, 2209, 2401, 2601, 2744, 2809, 3025, 3125, 3249, 3364, 3375, 3481

Offset: 1

Gus Wiseman, May 02 2019

nonn

The first term that is not a perfect power (A001597) is 11880, with prime indices {1,1,1,2,2,2,3,5} and prime signature {1,1,3,3}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A number n is totally abnormal iff (1) the prime indices of n do not cover an initial interval of positive integers, and either (2a) n is prime, or (2b) the prime exponents (or prime signature) of n forms a totally abnormal integer partition, or, equivalently to (2b), A181819(n) is totally abnormal.
The enumeration of totally abnormal integer partitions by sum is given by A325332.

			The sequence of terms together with their prime indices begins:
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    49: {4,4}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   196: {1,1,4,4}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   289: {7,7}
   343: {4,4,4}
   361: {8,8}
   441: {2,2,4,4}
   484: {1,1,5,5}
   529: {9,9}
   625: {3,3,3,3}
   676: {1,1,6,6}

Cf. A001597, A055932, A056239, A112798, A181819, A317089, A317090, A317246, A319152, A319810, A325332, A325370, A325372.

normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[10000],!PrimeQ[#]&&totabnQ[#]&]

cp's OEIS Frontend

A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

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A108572 Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.

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A325332 Number of totally abnormal integer partitions of n.

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A319811 Number of totally aperiodic integer partitions of n.

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A325373 Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.

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