A319616 -id:A319616 - cp's OEIS frontend

A340652 Number of non-isomorphic twice-balanced multiset partitions of weight n.

1, 1, 0, 2, 3, 6, 20, 65, 134, 482, 1562, 4974, 15466, 51768, 179055, 631737, 2216757, 7905325, 28768472, 106852116, 402255207, 1532029660, 5902839974, 23041880550, 91129833143, 364957188701, 1478719359501, 6058859894440, 25100003070184, 105123020009481, 445036528737301

Offset: 0

Gus Wiseman, Feb 07 2021

nonn

We define a multiset partition to be twice-balanced if all of the following are equal:
(1) the number of parts;
(2) the number of distinct vertices;
(3) the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 6 multiset partitions (empty column indicated by dot):
  {{1}}  .  {{1},{2,2}}  {{1,1},{2,2}}  {{1},{1},{2,3,3}}
            {{2},{1,2}}  {{1,2},{1,2}}  {{1},{2},{2,3,3}}
                         {{1,2},{2,2}}  {{1},{2},{3,3,3}}
                                        {{1},{3},{2,3,3}}
                                        {{2},{3},{1,2,3}}
                                        {{3},{3},{1,2,3}}

The co-balanced version is A319616.
The singly balanced version is A340600.
The cross-balanced version is A340651.
The version for factorizations is A340655.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A303975 counts distinct prime factors in prime indices.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A047993 counts balanced partitions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340653 counts balanced factorizations.
- A340657/A340656 list numbers with/without a twice-balanced factorization.
Cf. A001055, A001221, A001222, A007717, A316983, A320663, A339888, A340597, A340599, A340654.

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, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(m,n,k,y=1)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, y^t*subst(x*Polrev(K(q, t, min(k,n\t))), x, x^t)/t, O(x*x^n)))); s/m!}
seq(n)={Vec(1 + sum(k=1,n, polcoef(G(k,n,k,y) - G(k-1,n,k,y) - G(k,n,k-1,y) + G(k-1,n,k-1,y), k, y)))} \\ Andrew Howroyd, Jan 15 2024

a(11) onwards from Andrew Howroyd, Jan 15 2024

A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 2, 4, 11, 25, 75, 206, 650, 2049, 6895

Offset: 0

Gus Wiseman, Dec 26 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{2},{1,2}}  {{1,2,3,3}}
                               {{1,2,3,4}}
                               {{1},{1,2,2}}
                               {{1,2},{1,2}}
                               {{1,2},{2,2}}
                               {{1,3},{2,3}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
This is the connected case of A368098, ranks A368100.
Complement set-systems: A368409, connected case of A368094, ranks A367907.
For set-systems we have A368410, connected case of A368095, ranks A367906.
The complement is A368411, connected case of A368097, ranks A355529.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A140637, A302545, A316983, A317533, A319616, A367902, A367905, A368187.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute /@ Select[mpm[n],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A320801 Regular triangle read by rows where T(n,k) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 3, 6, 0, 1, 6, 10, 16, 0, 1, 6, 20, 30, 34, 0, 1, 9, 31, 75, 92, 90, 0, 1, 9, 45, 126, 246, 272, 211, 0, 1, 12, 60, 223, 501, 839, 823, 558, 0, 1, 12, 81, 324, 953, 1900, 2762, 2482, 1430, 0, 1, 15, 100, 491, 1611, 4033, 7120, 9299, 7629, 3908

Offset: 0

Gus Wiseman, Nov 09 2018

			Triangle begins:
   1
   0   1
   0   1   3
   0   1   3   6
   0   1   6  10  16
   0   1   6  20  30  34
   0   1   9  31  75  92  90
   0   1   9  45 126 246 272 211
   0   1  12  60 223 501 839 823 558

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A007716. Last column is A049311.

Cf. A316980, A316983, A317533, A318805, A319616, A319721, A320796, A320808, A321194.

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)={prod(j=1, #q, my(g=gcd(t, q[j]), e=(q[j]/g)); (1 - y^e + y^e/(1 - x^e) + O(x*x^k))^g) - 1}
G(n)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, substvec(K(q, t, n\t)/t, [x,y], [x^t,y^t])) + O(x*x^n))); s/n!}
T(n)=[Vecrev(p) | p<-Vec(G(n))]
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 16 2024

Offset corrected by Andrew Howroyd, Jan 16 2024

A321720 Number of non-normal (0,1) semi-magic squares with sum of entries equal to n.

Original entry on oeis.org

1, 1, 2, 6, 25, 120, 726, 5040, 40410, 362881, 3630840, 39916800, 479069574, 6227020800, 87181402140, 1307674370040, 20922977418841, 355687428096000, 6402388104196400, 121645100408832000, 2432903379962038320, 51090942171778378800, 1124000886592995642000, 25852016738884976640000

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Andrew Howroyd, Table of n, a(n) for n = 0..180
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A007016, A057151, A068313, A008300, A101370, A104602, A120732, A271103, A319056, A319616.

Cf. A321717, A321719, A321722, A321723.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(p) = p! for p prime as the squares are all permutation matrices of order p and a(n) >= n! for n > 1 (see comments in A321717 and A321719). - Chai Wah Wu, Jan 13 2019

a(n) = Sum_{d|n, d<=n/d} A008300(n/d, d) for n > 0. - Andrew Howroyd, Apr 11 2020

a(7) from Chai Wah Wu, Jan 13 2019
a(8)-a(15) from Chai Wah Wu, Jan 14 2019
a(16)-a(21) from Chai Wah Wu, Jan 16 2019
Terms a(22) and beyond from Andrew Howroyd, Apr 11 2020

A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 1, 1, 5, 1, 1, 3, 7, 1, 1, 1, 1, 4, 9, 12, 11, 1, 1, 1, 1, 4, 15, 1, 1, 13, 31, 1, 1, 5, 43, 22, 1, 1, 1, 1, 5, 22, 103, 30, 1, 1, 1, 1, 6, 106, 264, 42, 1, 1, 30, 383, 1, 1, 6, 56, 1, 1, 1, 1, 7, 45, 321, 2804, 1731, 77, 1

Offset: 1

Gus Wiseman, Nov 18 2018

Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d.
A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Triangle begins:
  1
  1 1
  1 1
  1 2 1
  1 1
  1 2 3 1
  1 1
  1 3 5 1
  1 5 1
  1 3 7 1
Inequivalent representatives of the T(10,3) = 7 semi-magic squares (zeros not shown):
  [2    ] [2    ] [2    ] [2    ] [2    ] [11   ] [11   ]
  [ 2   ] [ 2   ] [ 2   ] [ 11  ] [ 11  ] [11   ] [1 1  ]
  [  2  ] [  2  ] [  11 ] [ 11  ] [ 1 1 ] [  11 ] [ 1 1 ]
  [   2 ] [   11] [  1 1] [   11] [  1 1] [  1 1] [  1 1]
  [    2] [   11] [   11] [   11] [   11] [   11] [   11]

Andrew Howroyd, Table of n, a(n) for n = 1..207 (rows 1..50)
Wikipedia, Magic square

Row sums are A321721.
Cf. A006052, A007016, A007716, A027750, A057150, A120732, A271103, A319056, A319616.
Cf. A321718, A321719, A321722, A333733.

T(n,k) = A333733(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Apr 11 2020

a(28)-a(39) from Chai Wah Wu, Jan 16 2019
Terms a(40) and beyond from Andrew Howroyd, Apr 11 2020
Edited by Peter Munn, Mar 05 2025

A368411 Number of non-isomorphic connected multiset partitions of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 2, 6, 15, 50, 148, 509, 1725, 6218

Offset: 0

Gus Wiseman, Dec 26 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 15 multiset partitions:
  {{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},{1,1}}    {{1},{1},{1,1,1}}
                            {{1},{2},{1,2}}    {{1},{1,1},{1,1}}
                            {{2},{2},{1,2}}    {{1},{1},{1,2,2}}
                            {{1},{1},{1},{1}}  {{1},{1,2},{2,2}}
                                               {{1},{2},{1,2,2}}
                                               {{2},{1,2},{1,2}}
                                               {{2},{1,2},{2,2}}
                                               {{2},{2},{1,2,2}}
                                               {{3},{3},{1,2,3}}
                                               {{1},{1},{1},{1,1}}
                                               {{1},{2},{2},{1,2}}
                                               {{2},{2},{2},{1,2}}
                                               {{1},{1},{1},{1},{1}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A140638, connected case of A367867.
The complement for labeled graphs is A129271, connected case of A133686.
This is the connected case of A368097.
For set-systems we have A368409, connected case of A368094, ranks A367907.
Complement set-systems: A368410, connected case of A368095, ranks A367906.
The complement is A368412, connected case of A368098, ranks A368100.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A140637, A302545, A316983, A317533, A319616, A367903, A367905, A368187.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List /@ c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute /@ Select[mpm[n],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

A319766 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n whose dual is also a strict intersecting multiset partition.

Original entry on oeis.org

1, 1, 1, 4, 6, 14, 31, 64, 145, 324, 753

Offset: 0

Gus Wiseman, Sep 27 2018

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 partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.
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(5) = 14 multiset partitions:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{1,2},{2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1},{1,1,1,1}}
   {{1},{1,2,2,2}}
   {{2},{1,1,2,2}}
   {{2},{1,2,2,2}}
   {{2},{1,2,3,3}}
   {{1,1},{1,1,1}}
   {{1,1},{1,2,2}}
   {{1,2},{1,2,2}}
   {{1,2},{2,2,2}}
   {{2,2},{1,2,2}}
   {{2},{1,2},{2,2}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319765, A319767, A319768, A319769, A319773, A319774.

A319768 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n whose dual is a (not necessarily strict) intersecting multiset partition.

Original entry on oeis.org

1, 1, 2, 5, 11, 25, 63, 144, 364, 905, 2356

Offset: 0

Gus Wiseman, Sep 27 2018

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.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,2},{2,2}}
   {{1},{2},{1,2}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319765, A319766, A319767, A319769, A319773, A319774.

A319769 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n whose dual is also an intersecting set multipartition.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 16, 26, 38, 61

Offset: 0

Gus Wiseman, Sep 27 2018

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.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 7 set multipartitions:
1: {{1}}
2: {{1,2}}
   {{1},{1}}
3: {{1,2,3}}
   {{2},{1,2}}
   {{1},{1},{1}}
4: {{1,2,3,4}}
   {{3},{1,2,3}}
   {{1,2},{1,2}}
   {{2},{2},{1,2}}
   {{1},{1},{1},{1}}
5: {{1,2,3,4,5}}
   {{4},{1,2,3,4}}
   {{2,3},{1,2,3}}
   {{2},{1,2},{1,2}}
   {{3},{3},{1,2,3}}
   {{2},{2},{2},{1,2}}
   {{1},{1},{1},{1},{1}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319765, A319766, A319767, A319768, A319773, A319774.

A319773 Number of non-isomorphic intersecting set systems of weight n whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 2, 4, 5

Offset: 0

Gus Wiseman, Sep 27 2018

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.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			Non-isomorphic representatives of the a(1) = 1 through a(10) = 5 set systems:
1:  {{1}}
3:  {{2},{1,2}}
6:  {{3},{2,3},{1,2,3}}
    {{1,2},{1,3},{2,3}}
7:  {{1,3},{2,3},{1,2,3}}
8:  {{2,4},{3,4},{1,2,3,4}}
    {{3},{1,3},{2,3},{1,2,3}}
9:  {{1,2,4},{1,3,4},{2,3,4}}
    {{4},{2,4},{3,4},{1,2,3,4}}
    {{1,2},{1,3},{1,4},{2,3,4}}
    {{1,2},{1,3},{2,3},{1,2,3}}
10: {{4},{3,4},{2,3,4},{1,2,3,4}}
    {{4},{1,2,4},{1,3,4},{2,3,4}}
    {{1,2},{2,4},{1,3,4},{2,3,4}}
    {{1,4},{2,4},{3,4},{1,2,3,4}}
    {{2,3},{2,4},{3,4},{1,2,3,4}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319765, A319766, A319767, A319768, A319769, A319774.

cp's OEIS Frontend

A340652 Number of non-isomorphic twice-balanced multiset partitions of weight n.

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A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

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A320801 Regular triangle read by rows where T(n,k) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n.

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A321720 Number of non-normal (0,1) semi-magic squares with sum of entries equal to n.

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A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

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A368411 Number of non-isomorphic connected multiset partitions of weight n contradicting a strict version of the axiom of choice.

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Mathematica

A319766 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n whose dual is also a strict intersecting multiset partition.

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A319768 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n whose dual is a (not necessarily strict) intersecting multiset partition.

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Examples

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A319769 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n whose dual is also an intersecting set multipartition.

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