A049311 -id:A049311 - cp's OEIS frontend

A007716 Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.

1, 1, 4, 10, 33, 91, 298, 910, 3017, 9945, 34207, 119369, 429250, 1574224, 5916148, 22699830, 89003059, 356058540, 1453080087, 6044132794, 25612598436, 110503627621, 485161348047, 2166488899642, 9835209912767, 45370059225318, 212582817739535, 1011306624512711

Offset: 0

Colin Mallows

Also, the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations (cf. A120733).
This is a two-dimensional generalization of the partition function (A000041), which equals the number of length n vectors of nonnegative integers with sum n, equivalent under permutations. - Franklin T. Adams-Watters, Sep 19 2011
Also number of non-isomorphic multiset partitions of weight n. - Gus Wiseman, Sep 19 2011

			The 10 non-isomorphic multiset partitions of weight 3 are {{1, 1, 1}}, {{1, 1, 2}}, {{1, 2, 3}}, {{1}, {1, 1}}, {{1}, {1, 2}}, {{1}, {2, 2}}, {{1}, {2, 3}}, {{1}, {1}, {1}}, {{1}, {1}, {2}}, {{1}, {2}, {3}}.

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..30 from Seiichi Manyama)

Main diagonal of A318795.

Cf. A053307, A052365, A052366, A052367, A052372, A052373, A049311, A054688, A000041.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, 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_, k_] := SeriesCoefficient[1/Product[(1-x^LCM[p[[i]], q[[j]]])^GCD[ p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
M[m_, n_, k_] := Module[{s=0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
a[n_] := a[n] = M[n, n, n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 18}] (* Jean-François Alcover, May 03 2019, after Andrew Howroyd *)

\\ See A318795
a(n) = M(n,n,n); \\ Andrew Howroyd, Sep 03 2018

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))}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q,t,n)/t))), n)); s/n!} \\ Andrew Howroyd, Mar 29 2020

a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; x_1, x_2, ...) if we replace x_i with 1+x^i+x^(2*i)+x^(3*i)+x^(4*i)+..., where S_n X S_n is the Cartesian product of symmetric groups S_n of degree n. - Vladeta Jovovic, Mar 09 2000

More terms from Vladeta Jovovic, Jun 28 2000
a(19)-a(25) from Max Alekseyev, Jan 22 2010
a(0)=1 prepended by Alois P. Heinz, Feb 03 2019
a(26)-a(27) from Seiichi Manyama, Nov 23 2019

A283877 Number of non-isomorphic set-systems of weight n.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 44, 98, 244, 605, 1595, 4273, 12048, 34790, 104480, 322954, 1031556, 3389413, 11464454, 39820812, 141962355, 518663683, 1940341269, 7424565391, 29033121685, 115921101414, 472219204088, 1961177127371, 8298334192288, 35751364047676, 156736154469354

Offset: 0

Gus Wiseman, Mar 17 2017

nonn

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.

			Non-isomorphic representatives of the a(4)=9 set-systems are:
((1234)),
((1)(234)), ((3)(123)), ((12)(34)), ((13)(23)),
((1)(2)(12)), ((1)(2)(34)), ((1)(3)(23)),
((1)(2)(3)(4)).

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

Cf. A007716, A034691, A049311, A056156, A089259, A116540, A300913.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(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)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024

Euler transform of A300913.

a(0) = 1 prepended and terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 36, 61, 101, 166, 267, 433, 686, 1088, 1709, 2671, 4140, 6403, 9824, 15028, 22864, 34657, 52288, 78646, 117784, 175865, 261657, 388145, 573936, 846377, 1244475, 1825170, 2669776, 3895833, 5671127, 8236945, 11936594, 17261557, 24909756

Offset: 0

N. J. A. Sloane, Dec 23 2003

nonn

Number of complete set partitions of the integer partitions of n. This is the Euler transform of A000009. If we change the combstruct command from unlabeled to labeled, then we get A000258. - Thomas Wieder, Aug 01 2008

Number of set multipartitions (multisets of sets) of integer partitions of n. Also a(n) < A270995(n) for n>5. - Gus Wiseman, Apr 10 2016

			From _Gus Wiseman_, Oct 22 2018: (Start)
The a(6) = 22 set multipartitions of integer partitions of 6:
  (6)  (15)    (123)      (12)(12)      (1)(1)(1)(12)    (1)(1)(1)(1)(1)(1)
       (24)    (1)(14)    (1)(1)(13)    (1)(1)(1)(1)(2)
       (1)(5)  (1)(23)    (1)(2)(12)
       (2)(4)  (2)(13)    (1)(1)(1)(3)
       (3)(3)  (3)(12)    (1)(1)(2)(2)
               (1)(1)(4)
               (1)(2)(3)
               (2)(2)(2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row sums of A285229 and of A360763.

Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.

with(combstruct): A089259:= [H, {H=Set(T, card>=1), T=PowerSet (Sequence (Z, card>=1), card>=1)}, unlabeled]; 1, seq (count (A089259, size=j), j=1..16); # Thomas Wieder, Aug 01 2008
# second Maple program:
with(numtheory):
b:= proc(n, i)
      if n<0 or n>i*(i+1)/2 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n,i):= b(n-i, i-1) +b(n, i-1)
      fi
    end:
a:= proc(n) option remember; `if` (n=0, 1,
       add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 11 2011

max = 40; CoefficientList[Series[Product[1/(1-x^m)^PartitionsQ[m], {m, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 24 2014 *)
b[n_, i_] := b[n, i] = Which[n<0 || n>i*(i+1)/2, 0, n == 0, 1, i<1, 0, True, b[n-i, i-1] + b[n, i-1]]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d* b[d, d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat([1], EulerT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)))} \\ Andrew Howroyd, Oct 26 2018

A316980 Number of non-isomorphic strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 3, 8, 23, 63, 197, 588, 1892, 6140, 20734, 71472, 254090, 923900, 3446572, 13149295, 51316445, 204556612, 832467052, 3455533022, 14621598811, 63023667027, 276559371189, 1234802595648, 5606647482646, 25875459311317, 121324797470067, 577692044073205

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows (or alternatively, with no equal columns).

Also the number of non-isomorphic multiset partitions of weight n with no equivalent vertices. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 8 multiset partitions with no equivalent vertices (first column) and with no equal blocks (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(1)(1) <-> (123)
  (1)(2)(2) <-> (1)(23)
  (1)(2)(3) <-> (1)(2)(3)

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

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091.
Cf. A316974, A316978, A316979, A316981, A316983.
Cf. A049311, A059201, A319558, A319560, A319567.

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))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(p=sum(t=1, n, subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*polcoef(exp(p-subst(p,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 21 2023

Euler transform of A319557. - Gus Wiseman, Sep 23 2018

a(7)-a(10) from Gus Wiseman, Sep 23 2018

Terms a(11) and beyond from Andrew Howroyd, Jan 19 2023

A116540 Number of zero-one matrices with n ones and no zero rows or columns, up to permutation of rows.

Original entry on oeis.org

1, 1, 3, 10, 41, 192, 1025, 6087, 39754, 282241, 2159916, 17691161, 154192692, 1423127819, 13851559475, 141670442163, 1517880400352, 16989834719706, 198191448685735, 2404300796114642, 30273340418567819, 394948562421362392, 5330161943597341380, 74307324695105372519

Offset: 0

Vladeta Jovovic, Mar 27 2006

nonn

Also number of normal set multipartitions of weight n. These are defined as multisets of sets that together partition a normal multiset of weight n, where a multiset is normal if it spans an initial interval of positive integers. Set multipartitions are involved in the expansion of elementary symmetric functions in terms of augmented monomial symmetric functions. - Gus Wiseman, Oct 22 2015

			The a(3) = 10 normal set multipartitions are: {1,1,1}, {1,12}, {1,1,2}, {2,12}, {1,2,2}, {123}, {1,23}, {2,13}, {3,12}, {1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..230
P. J. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, arXiv:math/0510155 [math.CO], 2005-2006.
M. Klazar, Extremal problems for ordered hypergraphs, arXiv:math/0305048 [math.CO], 2003.
Gus Wiseman, Four symmetric function identities

Cf. A049311, A101370.

Row sums of A327117.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j,
      min(n-i*j, i-1), k)*binomial(binomial(k, i)+j-1, j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..24);  # Alois P. Heinz, Sep 13 2019

MSOSA[s_List] :=
  MSOSA[s] = If[Length[s] === 0, {{}}, Module[{sbs, fms},
     sbs = Rest[Subsets[Union[s]]];
     fms =
      Function[r,
        Append[#, r] & /@
         MSOSA[Fold[DeleteCases[#1, #2, {1}, 1] &, s, r]]] /@ sbs;
     Select[Join @@ fms, OrderedQ]
     ]];
mmallnorm[n_Integer] :=
  Function[s, Array[Count[s, y_ /; y <= #] + 1 &, n]] /@
   Subsets[Range[n - 1] + 1];
Array[Plus @@ Length /@ MSOSA /@ mmallnorm[#] &, 9]
(* Gus Wiseman, Oct 22 2015 *)

R(n, k)={Vec(-1 + 1/prod(j=1, k, (1 - x^j + O(x*x^n))^binomial(k, j) ))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

a(0)=1 prepended and more terms added by Alois P. Heinz, Sep 13 2019

A318360 Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 5, 3, 2, 1, 6, 1, 2, 3, 15, 1, 9, 1, 6, 3, 2, 1, 21, 4, 2, 16, 6, 1, 10, 1, 52, 3, 2, 4, 35, 1, 2, 3, 22, 1, 10, 1, 6, 19, 2, 1, 83, 5, 13, 3, 6, 1, 66, 4, 22, 3, 2, 1, 41, 1, 2, 20, 203, 4, 10, 1, 6, 3, 14, 1, 153, 1, 2, 26, 6, 5, 10, 1

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			The a(12) = 6 set multipartitions of {1,1,2,3}:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A001055, A007716, A049311, A116540, A181821, A188392, A255906.

Cf. A318283, A318284, A318286, A318361, A318362, A318369.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,80}]

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}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i,2], j, primepi(f[i,1]))))}
count(sig)={my(n=vecsum(sig), s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + O(x*x^n))); s+=prod(i=1, #sig, polcoef(q,sig[i]))*permcount(p)); s/n!}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s<=2, if(#s==1, 1, min(s[1],s[2])+1), count(sig(n))))} \\ Andrew Howroyd, Dec 10 2018

a(n) = A050320(A181821(n)).
From Andrew Howroyd, Dec 10 2018:(Start)
a(p) = 1 for prime(p).
a(prime(i)*prime(j)) = min(i,j) + 1.
a(prime(n)^k) = A188392(n,k). (End)

A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

Original entry on oeis.org

1, 1, 4, 6, 17, 14, 66, 30, 189, 222, 550, 112, 4696, 202, 5612, 30914, 63219, 594, 453125, 980, 3602695, 5914580, 1169348, 2510, 299083307, 232988061, 23248212, 2669116433, 14829762423, 9130, 170677509317, 13684, 1724710753084, 2199418340875, 14184712185, 38316098104262

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.
Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018
From Gus Wiseman, Oct 11 2018: (Start)
Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{1},{2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{3,4}}
  {{1},{1},{1,1}}
  {{1},{1},{2,2}}
  {{1},{2},{1,2}}
  {{1},{2},{3,4}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{3},{4}}
(End)

			Non-isomorphic representatives of the a(4) = 17 multiset partitions:
  {{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}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{2},{2}}
  {{1},{2},{3},{3}}
  {{1},{2},{3},{4}}

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

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951.

Cf. A064573, A279787, A305551, A319616, A319056, A331485.

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, 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];
K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
a[n_] := a[n] = If[n==0, 1, If[PrimeQ[n], 2 PartitionsP[n], Sum[ RowSumMats[ n/d, n, d], {d, Divisors[n]}]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 07 2019, after Andrew Howroyd *)

\\ See A318951 for RowSumMats.
a(n)={sumdiv(n,d,RowSumMats(n/d,n,d))} \\ Andrew Howroyd, Sep 05 2018

For p prime, a(p) = 2*A000041(p).

a(n) = Sum_{d|n} A331485(n/d, d). - Andrew Howroyd, Feb 09 2020

Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018

A116539 Number of zero-one matrices with n ones and no zero rows or columns and with distinct rows, up to permutation of rows.

Original entry on oeis.org

1, 1, 2, 7, 28, 134, 729, 4408, 29256, 210710, 1633107, 13528646, 119117240, 1109528752, 10889570768, 112226155225, 1210829041710, 13640416024410, 160069458445202, 1952602490538038, 24712910192430620, 323964329622503527, 4391974577299578248, 61488854148194151940

Offset: 0

Vladeta Jovovic, Mar 27 2006

nonn

Also the number of labeled hypergraphs spanning an initial interval of positive integers with edge-sizes summing to n. - Gus Wiseman, Dec 18 2018

			From _Gus Wiseman_, Dec 18 2018: (Start)
The a(3) = 7 edge-sets:
    {{1,2,3}}
   {{1},{1,2}}
   {{2},{1,2}}
   {{1},{2,3}}
   {{2},{1,3}}
   {{3},{1,2}}
  {{1},{2},{3}}
Inequivalent representatives of the a(4) = 28 0-1 matrices:
  [1111]
.
  [100][1000][010][0100][001][0010][0001][110][110][1100][101][1010][1001]
  [111][0111][111][1011][111][1101][1110][101][011][0011][011][0101][0110]
.
  [10][100][100][1000][100][100][1000][1000][010][010][0100][0100][0010]
  [01][010][010][0100][001][001][0010][0001][001][001][0010][0001][0001]
  [11][101][011][0011][110][011][0101][0110][110][101][1001][1010][1100]
.
  [1000]
  [0100]
  [0010]
  [0001]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
P. J. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes , arXiv:math/0510155 [math.CO], 2005-2006.
M. Klazar, Extremal problems for ordered hypergraphs, arXiv:math/0305048 [math.CO], 2003.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763
Cf. A049311, A058891, A101370, A255906, A283877, A306021, A319190.
Row sums of A326914 and of A326962.

b:= proc(n, i, k) b(n, i, k):=`if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j,
      min(n-i*j, i-1), k)*binomial(binomial(k, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 13 2019

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k]*Binomial[Binomial[k, i], j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

a(0)=1 prepended and more terms added by Alois P. Heinz, Sep 13 2019

A319056 Number of non-isomorphic multiset partitions of weight n in which (1) all parts have the same size and (2) each vertex appears the same number of times.

Original entry on oeis.org

1, 1, 4, 4, 10, 4, 21, 4, 26, 13, 28, 4, 128, 4, 39, 84, 150, 4, 358, 4, 956, 513, 86, 4, 12549, 1864, 134, 9582, 52366, 4, 301086, 4, 1042038, 407140, 336, 4690369, 61738312, 4, 532, 28011397, 2674943885, 4, 819150246, 4, 54904825372, 65666759973, 1303, 4, 4319823776760

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

a(p) = 4 for p prime. - Charlie Neder, Oct 15 2018

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 21 multiset partitions:
  (1)  (11)    (111)      (1111)        (11111)          (111111)
       (12)    (123)      (1122)        (12345)          (111222)
       (1)(1)  (1)(1)(1)  (1234)        (1)(1)(1)(1)(1)  (112233)
       (1)(2)  (1)(2)(3)  (11)(11)      (1)(2)(3)(4)(5)  (123456)
                          (11)(22)                       (111)(111)
                          (12)(12)                       (111)(222)
                          (12)(34)                       (112)(122)
                          (1)(1)(1)(1)                   (112)(233)
                          (1)(1)(2)(2)                   (123)(123)
                          (1)(2)(3)(4)                   (123)(456)
                                                         (11)(11)(11)
                                                         (11)(12)(22)
                                                         (11)(22)(33)
                                                         (11)(23)(23)
                                                         (12)(12)(12)
                                                         (12)(13)(23)
                                                         (12)(34)(56)
                                                         (1)(1)(1)(1)(1)(1)
                                                         (1)(1)(1)(2)(2)(2)
                                                         (1)(1)(2)(2)(3)(3)
                                                         (1)(2)(3)(4)(5)(6)

Andrew Howroyd, Table of n, a(n) for n = 0..75

Row sums of A322789.

Cf. A007716, A049311, A064573, A279787, A283877, A306017, A316980, A316981, A316983, A319616.

Terms a(12) and beyond from Andrew Howroyd, Feb 03 2022

A293606 Number of unlabeled antichains of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 20, 33, 72, 139

Offset: 0

Gus Wiseman, Oct 13 2017

An antichain is a finite set of finite nonempty sets, none of which is a subset of any other. The weight of an antichain is the sum of cardinalities of its elements.
From Gus Wiseman, Aug 15 2019: (Start)
Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n where every vertex is the unique common element of some subset of the edges. For example, the a(1) = 1 through a(6) = 20 set multipartitions are:
  {1}  {1}{1}  {1}{1}{1}  {1}{2}{12}    {1}{2}{2}{12}    {12}{13}{23}
       {1}{2}  {1}{2}{2}  {1}{1}{1}{1}  {1}{2}{3}{23}    {1}{2}{12}{12}
               {1}{2}{3}  {1}{1}{2}{2}  {1}{1}{1}{1}{1}  {1}{2}{13}{23}
                          {1}{2}{2}{2}  {1}{1}{2}{2}{2}  {1}{2}{3}{123}
                          {1}{2}{3}{3}  {1}{2}{2}{2}{2}  {1}{1}{2}{2}{12}
                          {1}{2}{3}{4}  {1}{2}{2}{3}{3}  {1}{1}{2}{3}{23}
                                        {1}{2}{3}{3}{3}  {1}{2}{2}{2}{12}
                                        {1}{2}{3}{4}{4}  {1}{2}{3}{3}{23}
                                        {1}{2}{3}{4}{5}  {1}{2}{3}{4}{34}
                                                         {1}{1}{1}{1}{1}{1}
                                                         {1}{1}{1}{2}{2}{2}
                                                         {1}{1}{2}{2}{2}{2}
                                                         {1}{1}{2}{2}{3}{3}
                                                         {1}{2}{2}{2}{2}{2}
                                                         {1}{2}{2}{3}{3}{3}
                                                         {1}{2}{3}{3}{3}{3}
                                                         {1}{2}{3}{3}{4}{4}
                                                         {1}{2}{3}{4}{4}{4}
                                                         {1}{2}{3}{4}{5}{5}
                                                         {1}{2}{3}{4}{5}{6}
(End)

			Non-isomorphic representatives of the a(5) = 9 antichains are:
((12345)),
((1)(2345)), ((12)(134)), ((12)(345)),
((1)(2)(345)), ((1)(23)(45)), ((2)(13)(14)),
((1)(2)(3)(45)),
((1)(2)(3)(4)(5)).

Cf. A006126, A006602, A007411, A007716, A048143, A049311, A283877, A293607.

Cf. A000372, A000612, A003182, A014466, A055621, A293993, A326704, A326972.

Euler transform of A293607.

cp's OEIS Frontend

A007716 Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.

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A283877 Number of non-isomorphic set-systems of weight n.

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A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

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A316980 Number of non-isomorphic strict multiset partitions of weight n.

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A116540 Number of zero-one matrices with n ones and no zero rows or columns, up to permutation of rows.

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A318360 Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

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A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

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