A255906
Number of collections of nonempty multisets with a total of n objects having color set {1,...,k} for some k<=n.
Original entry on oeis.org
1, 1, 4, 16, 76, 400, 2356, 15200, 106644, 806320, 6526580, 56231024, 513207740, 4941362512, 50013751812, 530481210672, 5880285873060, 67954587978448, 816935340368068, 10196643652651664, 131904973822724540, 1765645473517011568, 24420203895517396180
Offset: 0
a(0) = 1: {}.
a(1) = 1: {{1}}.
a(2) = 4: {{1},{1}}, {{1,1}}, {{1},{2}}, {{1,2}}.
a(3) = 16: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{1},{2}}, {{1},{2},{2}}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}, {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}.
-
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]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Join@@mps/@allnorm[n]],{n,6}] (* Gus Wiseman, Jul 30 2018 *)
-
R(n, k)={Vec(-1 + 1/prod(j=1, n, (1 - x^j + O(x*x^n))^binomial(k+j-1, 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
A317757
Number of non-isomorphic multiset partitions of size n such that the blocks have empty intersection.
Original entry on oeis.org
1, 0, 1, 4, 17, 56, 205, 690, 2446, 8506, 30429, 109449, 402486, 1501424, 5714194, 22132604, 87383864, 351373406, 1439320606, 6003166059, 25488902820, 110125079184, 483987225922, 2162799298162, 9823464989574, 45332196378784, 212459227340403, 1010898241558627, 4881398739414159
Offset: 0
Non-isomorphic representatives of the a(4) = 17 multiset partitions:
{1}{234},{2}{111},{2}{113},{11}{22},{11}{23},{12}{34},
{1}{1}{22},{1}{1}{23},{1}{2}{11},{1}{2}{12},{1}{2}{13},{1}{2}{34},{2}{3}{11},
{1}{1}{1}{2},{1}{1}{2}{2},{1}{1}{2}{3},{1}{2}{3}{4}.
-
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];
sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,strnorm[n]}]]],{n,6}]
-
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))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={my(s=0); forpart(q=n, my(f=prod(i=1, #q, 1 - x^q[i]), u=R(q,n)); s+=permcount(q)*sum(k=0, n, my(c=polcoef(f,k)); if(c, c*polcoef(exp(sum(t=1, n\(k+1), x^(t*k)*u[t], O(x*x^n) ))/if(k,1-x^k,1), n))) ); s/n!} \\ Andrew Howroyd, May 30 2023
a(0)=1 prepended and terms a(11) and beyond from
Andrew Howroyd, May 30 2023
A317752
Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.
Original entry on oeis.org
0, 1, 8, 49, 305, 1984, 13686, 100124, 776885, 6386677, 55532358, 509549386, 4921352952, 49899820572, 529807799836, 5876162077537, 67928460444139, 816764249684450, 10195486840926032, 131896905499007474, 1765587483656124106, 24419774819813602870
Offset: 1
The a(3) = 8 multiset partitions with empty intersection:
{{2},{1,1}}
{{1},{2,2}}
{{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{1},{2}}
{{1},{2},{2}}
{{1},{2},{3}}
-
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]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,allnorm[n]}]],{n,6}]
-
P(n,k)={1/prod(i=1, n, (1 - x^i*y + O(x*x^n))^binomial(k+i-1, k-1))}
R(n,k)={my(p=P(n,k), q=p/(1-y+O(y*y^n))); Vec(sum(i=2, n, polcoef(p,i,y) + polcoef(q,i,y)*sum(j=1, n\i, (-1)^j*binomial(k,j)*x^(i*j))), -n)}
seq(n)={sum(k=2, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Feb 05 2021
A317755
Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.
Original entry on oeis.org
0, 1, 6, 30, 130, 629, 2930, 15019, 78224, 438626, 2548481
Offset: 1
The a(3) = 6 strongly normal multiset partitions with empty intersection:
{{2},{1,1}}
{{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{1},{2}}
{{1},{2},{3}}
-
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[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,strnorm[n]}]],{n,6}]
A317077
Number of connected multiset partitions of normal multisets of size n.
Original entry on oeis.org
1, 1, 3, 8, 28, 110, 519, 2749, 16317, 106425, 755425, 5781956, 47384170, 413331955, 3818838624, 37213866876, 381108145231, 4088785729738, 45829237977692, 535340785268513, 6502943193997922, 81984445333355812, 1070848034863526547, 14467833457108560375, 201894571410270034773
Offset: 0
The a(3) = 8 connected multiset partitions are (111), (1)(11), (1)(1)(1), (122), (2)(12), (112), (1)(12), (123).
-
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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Length/@Table[Join@@Table[Select[mps[m],Length[csm[#]]==1&],{m,allnorm[n]}],{n,8}]
-
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k]));u}
seq(n)={my(u=vector(n, k, x*Ser(EulerT(vector(n,i,binomial(i+k-1,i)))))); Vec(1+vecsum(Connected(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*u[i])))))} \\ Andrew Howroyd, Jan 16 2023
A317074
Number of antichains of multisets with multiset-join a strongly normal multiset of size n.
Original entry on oeis.org
1, 1, 3, 13, 148, 7685
Offset: 0
The a(3) = 13 antichains of multisets:
(111),
(112), (11)(12), (2)(11),
(123), (13)(23), (12)(23), (12)(13), (12)(13)(23), (3)(12), (2)(13), (1)(23), (1)(2)(3).
-
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],multijoin@@#==m&];
Table[Length[Join@@Table[auu[m],{m,strnorm[n]}]],{n,5}]
A317075
Number of connected antichains of multisets with multiset-join a normal multiset of size n.
Original entry on oeis.org
1, 1, 2, 10, 147, 8998
Offset: 0
The a(3) = 10 connected antichains of multisets:
(111),
(122), (12)(22),
(112), (11)(12),
(123), (13)(23), (12)(23), (12)(13), (12)(13)(23).
-
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],And[multijoin@@#==m,Length[csm[#]]==1]&];
Table[Length[Join@@Table[cuu[m],{m,allnorm[n]}]],{n,5}]
A317080
Number of unlabeled connected antichains of multisets with multiset-join a multiset of size n.
Original entry on oeis.org
1, 1, 2, 6, 34, 392
Offset: 0
Non-isomorphic representatives of the a(3) = 6 connected antichains of multisets:
(111),
(122), (12)(22),
(123), (13)(23), (12)(13)(23).
-
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}]
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
sysnorm[m_]:=First[Sort[sysnorm[m,1]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],And[multijoin@@#==m,Length[csm[#]]==1]&];
Table[Length[Union[sysnorm/@Join@@Table[cuu[m],{m,strnorm[n]}]]],{n,5}]
A317079
Number of unlabeled antichains of multisets with multiset-join a multiset of size n.
Original entry on oeis.org
1, 1, 3, 9, 46, 450
Offset: 0
Non-isomorphic representatives of the a(3) = 9 antichains of multisets:
(111),
(122), (1)(22), (12)(22),
(123), (1)(23), (13)(23), (1)(2)(3), (12)(13)(23).
-
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}]
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],multijoin@@#==m&];
sysnorm[m_]:=First[Sort[sysnorm[m,1]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Join@@Table[auu[m],{m,strnorm[n]}]]],{n,5}]
Showing 1-9 of 9 results.
Comments