A049311 -id:A049311 - cp's OEIS frontend

A318362 Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 5, 1, 2, 3, 5, 1, 7, 1, 5, 3, 2, 1, 9, 4, 2, 8, 5, 1, 10

Offset: 1

Gus Wiseman, Aug 24 2018

			Non-isomorphic representatives of the a(12) = 5 set multipartitions of {1,1,2,3}:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{1},{2},{3}}

Cf. A007716, A049311, A116540, A181821, A317791.

Cf. A318283, A318285, A318287, A318360, A318361, A318369, A318371.

a(n) = A318369(A181821(n)).

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

A321446 Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows and columns.

Original entry on oeis.org

1, 1, 2, 10, 72, 624, 6522, 80178, 1129368, 17917032, 316108752, 6138887616, 130120838400, 2989026225696, 73964789192400, 1961487062520720, 55495429438186920, 1668498596700706440, 53122020640948010640, 1785467619718933936560, 63175132023953553400440

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

			The a(3) = 10 matrices:
  [1 1] [1 1] [1 0] [0 1]
  [1 0] [0 1] [1 1] [1 1]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

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

Cf. A000612, A007716, A049311, A101370, A120733, A135589, A283877, A316980, A319559, A321515, A321586, A321587, A321588, A369285.

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/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#],UnsameQ@@Transpose[prs2mat[#]]]&]],{n,6}]

\\ Q(m, n, wf) defined in A321588.
seq(n)={my(R=vectorv(n,m,Q(m,n,w->1 + y^w + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum(vecsum(R)))} \\ Andrew Howroyd, Jan 24 2024

a(7) onwards from Andrew Howroyd, Jan 20 2024

A330783 Number of set multipartitions (multisets of sets) of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 3, 8, 27, 94, 385, 1673, 8079, 41614, 231447, 1364697, 8559575, 56544465, 393485452, 2867908008, 21869757215, 173848026202, 1438593095272, 12360614782433, 110119783919367, 1015289796603359, 9674959683612989, 95147388659652754, 964559157655032720, 10067421615492769230

Offset: 0

Gus Wiseman, Jan 02 2020

nonn

The (weakly) normal version is A116540.

			The a(1) = 1 through a(3) = 8 set multipartitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
The a(4) = 27 set multipartitions:
  {{1},{1},{1},{1}}  {{1},{1},{1,2}}  {{1},{1,2,3}}  {{1,2,3,4}}
  {{1},{1},{1},{2}}  {{1},{1},{2,3}}  {{1,2},{1,2}}
  {{1},{1},{2},{2}}  {{1},{2},{1,2}}  {{1,2},{1,3}}
  {{1},{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{2,3,4}}
  {{1},{2},{3},{4}}  {{1},{2},{3,4}}  {{1,2},{3,4}}
                     {{1},{3},{1,2}}  {{1,3},{2,4}}
                     {{1},{3},{2,4}}  {{1,4},{2,3}}
                     {{1},{4},{2,3}}  {{2},{1,3,4}}
                     {{2},{3},{1,4}}  {{3},{1,2,4}}
                     {{2},{4},{1,3}}  {{4},{1,2,3}}
                     {{3},{4},{1,2}}

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

Allowing edges to be multisets gives is A035310.
The strict case is A318402.
The constant case is A000005.
The (weakly) normal version is A116540.
Unlabeled set multipartitions are A049311.
Set multipartitions of prime indices are A050320.
Set multipartitions of integer partitions are A089259.
Cf. A001055, A047968, A255906, A269134, A283877, A296119, A317775, A318360, A318362, A330625, A330628.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],And@@UnsameQ@@@#&]],{n,0,5}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n)))/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Dec 30 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

A052371 Triangle T(n,k) of n X n binary matrices with k=0...n^2 ones up to row and column permutations.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 6, 7, 7, 6, 3, 1, 1, 1, 1, 3, 6, 16, 21, 39, 44, 55, 44, 39, 21, 16, 6, 3, 1, 1, 1, 1, 3, 6, 16, 34, 69, 130, 234, 367, 527, 669, 755, 755, 669, 527, 367, 234, 130, 69, 34, 16, 6, 3, 1, 1

Offset: 0

Vladeta Jovovic, Mar 08 2000

			Triangle begins:
  1;
  1, 1;
  1, 1, 3, 1, 1;
  1, 1, 3, 6, 7, 7, 6, 3, 1, 1;
  1, 1, 3, 6, 16, 21, 39, 44, 55, 44, 39, 21, 16, 6, 3, 1, 1;
  ...
(the last block giving the numbers of 4 X 4 binary matrices with k=0..16 ones up to row and column permutations).

Andrew Howroyd, Table of n, a(n) for n = 0..2890 (rows n=0..20)

Rows 6..8 are A052370, A053304, A053305.
Row sums are A002724.
Cf. A049311.

permcount[v_] := Module[{m = 1, s = 0, t, i, k = 0}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
c[p_, q_] := Product[(1 + x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {i, 1, Length[p]}, {j, 1, Length[q]}];
row[n_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q], {q, IntegerPartitions[n]}], {p, IntegerPartitions[n]}]; CoefficientList[ s/(n!^2), x]]
row /@ Range[0, 5] // Flatten (* Jean-François Alcover, Sep 22 2019, after Andrew Howroyd *)

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}
c(p, q)={prod(i=1, #p, prod(j=1, #q, (1 + x^lcm(p[i], q[j]))^gcd(p[i], q[j])))}
row(n)={my(s=0); forpart(p=n, forpart(q=n, s+=permcount(p) * permcount(q) * c(p, q))); Vec(s/(n!^2))}
for(n=1, 5, print(row(n))) \\ Andrew Howroyd, Nov 14 2018

a(0)=1 prepended by Andrew Howroyd, Nov 14 2018

A306008 Number of non-isomorphic intersecting set-systems of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 7, 10, 21, 39, 78

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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

Cf. A007716, A034691, A048143, A049311, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306007.

A318369 Number of non-isomorphic set multipartitions (multisets of sets) of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 5, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			Non-isomorphic representatives of the a(180) = 7 set multipartitions of {1,1,2,2,3}:
  {{1,2},{1,2,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{3},{1,2},{1,2}}
  {{1},{1},{2},{2,3}}
  {{1},{2},{3},{1,2}}
  {{1},{1},{2},{2},{3}}

Cf. A001055, A007716, A056239, A112798, A049311, A116540, A317791, A318360, A318361.

A321587 Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows.

Original entry on oeis.org

1, 1, 3, 17, 129, 1227, 14123, 190265, 2934359, 50975647, 984801759, 20941104299, 486007744671, 12223797601887, 331190083773701, 9616356919931711, 297887922137531747, 9805965265937326129, 341827167387114704421, 12579123760272833723975, 487315396984696657840761

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

Also number of colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i have i distinct colors in increasing order. a(3) = 17: 2ab1a, 2ab1b, 1a2ab, 1b2ab, 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a. - Alois P. Heinz, Sep 17 2019

			The a(3) = 17 matrices:
  [1 1 1]
.
  [1 1] [1 1] [1 1 0] [1 0 1] [1 0] [1 0 0] [0 1 1] [0 1] [0 1 0] [0 0 1]
  [1 0] [0 1] [0 0 1] [0 1 0] [1 1] [0 1 1] [1 0 0] [1 1] [1 0 1] [1 1 0]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A007716, A049311, A101370, A120733, A283877, A316980, A321446, A321515.

Row sums of A327583, A327584.

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 16 2019

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/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#]]&]],{n,5}]

a(n) ~ c * d^n * n!, where d = 1.938593839617140963759657977... and c = 0.350862127201784401195038... - Vaclav Kotesovec, Feb 05 2022

a(7)-a(20) from Alois P. Heinz, Sep 16 2019

A321735 Number of (0,1)-matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.

Original entry on oeis.org

1, 1, 2, 7, 30, 153, 939, 6653, 53743, 486576

Offset: 0

Gus Wiseman, Nov 18 2018

			The a(3) = 7 matrices:
  [1 1]
  [1 0]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Cf. A000700, A007016, A049311, A054976, A057151, A104602, A320451, A321719, A321723, A321732, A321733, A321736, A321739.

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/@#]==Range[Max@@First/@#]==Union[Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

Let c(y) be the coefficient of m(y) in e(y), where m is monomial symmetric functions and e is elementary symmetric functions. Then a(n) = Sum_{|y| = n} c(y).

A330677 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth whose leaves (which are multisets of atoms) are sets.

Original entry on oeis.org

1, 1, 1, 2, 11, 81, 859

Offset: 0

Gus Wiseman, Dec 30 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 11 multisystems:
  {}  {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}
                  {{1},{2,3}}  {{{1}},{{1},{2,3}}}
                               {{{1,2}},{{1},{1}}}
                               {{{1}},{{2},{1,2}}}
                               {{{1,2}},{{1},{2}}}
                               {{{1}},{{2},{1,3}}}
                               {{{1,2}},{{1},{3}}}
                               {{{1}},{{2},{3,4}}}
                               {{{1,2}},{{3},{4}}}
                               {{{2}},{{1},{1,3}}}
                               {{{2,3}},{{1},{1}}}

The version with all distinct atoms is A000111.
Non-isomorphic set multipartitions are A049311.
The (non-maximal) tree version is A330626.
Allowing leaves to be multisets gives A330663.
The case with prescribed degrees is A330664.
The version allowing all depths is A330668.
Cf. A000669, A001678, A004114, A005121, A007716, A141268, A283877, A306186, A330465, A330470, A330624.

cp's OEIS Frontend

A318362 Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

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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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A321446 Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows and columns.

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Mathematica

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A330783 Number of set multipartitions (multisets of sets) of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

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A052371 Triangle T(n,k) of n X n binary matrices with k=0...n^2 ones up to row and column permutations.

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Mathematica

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A306008 Number of non-isomorphic intersecting set-systems of weight n with no singletons.

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A318369 Number of non-isomorphic set multipartitions (multisets of sets) of the multiset of prime indices of n.

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A321587 Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows.

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Maple

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A321735 Number of (0,1)-matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.

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