A317658 -id:A317658 - cp's OEIS frontend

A277996 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

1, 1, 2, 5, 13, 36, 102, 299, 892, 2713, 8364, 26108, 82310, 261804, 838961, 2706336, 8780725, 28636157, 93818641, 308641277, 1019140129, 3376604826, 11221805968, 37399728251, 124967677989, 418564867751, 1405030366113, 4726036692421, 15927027834163, 53770343259613

Offset: 1

Gus Wiseman, Dec 24 2016

nonn

Also the number of distinct orderless Mathematica expressions with one atom and n positions.

			The a(5)=13 Mathematica expressions are:
x[x,x,x]
x[x,x][]   x[x][x]   x[][x,x]  x[x,x[]]  x[x[x]]
x[x][][]   x[][x][]  x[][][x]  x[x[]][]  x[][x[]]  x[x[][]]
x[][][][]

Andrew Howroyd, Table of n, a(n) for n = 1..200
Mathematica Reference, Orderless.

Cf. A000108, A001003, A005043, A052893, A279944, A280000, A317658.

multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
a[n_]:=a[n]=If[n===1,1,Sum[a[k]*Sum[Product[multing[a[First[s]],Length[s]],{s,Split[p]}],{p,IntegerPartitions[n-k-1]}],{k,1,n-1}]];
Array[a,30]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} A(x^k)/k)).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + (Sum_{n>=1} a(n)*x^n) * Product_{n>=1} 1/(1 - x^n)^a(n)). (End)

A317654 Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.

Original entry on oeis.org

1, 3, 26, 375, 6696, 159837, 4389226, 144915350, 5377002075, 227624621051, 10632808475596, 550932945236121, 31062550998284221, 1907051034025848314, 126052420069459211076, 8956882232940915920404, 679298518935625486287703, 54868537321267493152151502, 4696952405203792017289469056

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.

			The a(3) = 26 free pure symmetric multifunctions:
1[1[1]], 1[1,1], 1[1][1],
1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1],
1[2[3]], 1[3[2]], 1[2,3], 2[1[3]], 2[3[1]], 2[1,3], 3[1[2]], 3[2[1]], 3[1,2], 1[2][3], 2[1][3], 1[3][2], 3[1][2], 2[3][1], 3[2][1].

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317655, A317656, A317658.

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]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0,{},If[Length[m]==1,{First[m]},Join@@Cases[Union[Table[PR[m[[s]],m[[Complement[Range[Length[m]],s]]]],{s,Take[Subsets[Range[Length[m]]],{2,-2}]}]],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h],Union[Sort/@Tuples[exprUsing/@p]]}],{p,mps[g]}]]]];
got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
Table[Sum[Length[exprUsing[got[y]]],{y,IntegerPartitions[n]}],{n,6}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + p*(sExp(p)-1)); p}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Jan 01 2021

Terms a(8) and beyond from Andrew Howroyd, Jan 01 2021

A316112 Number of leaves in the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			e(21025) = o[o[o]][o] has 4 leaves so a(21025) = 4.

Cf. A007916, A052409, A052410, A109129, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.

nn=1000;
radQ[n_]:=If[n==1,False,GCD@@FactorInteger[n][[All,2]]==1];
rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},a[radPi[Power[n,1/g]]]+Sum[a[PrimePi[pr[[1]]]]*pr[[2]],{pr,If[g==1,{},FactorInteger[g]]}]]];
Table[a[n],{n,100}]

a(rad(x)^(prime(y_1) * ... * prime(y_k))) = a(x) + a(y_1) + ... + a(y_k) where rad = A007916.

A317056 Depth of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 3, 4, 6, 1, 4, 4, 5, 7, 2, 5, 5, 6, 3, 8, 2, 3, 6, 6, 7, 3, 4, 9, 3, 2, 4, 7, 7, 8, 4, 5, 10, 4, 3, 5, 8, 8, 4, 9, 5, 6, 11, 5, 4, 6, 9, 9, 5, 10, 6, 7, 12, 2, 6, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 7, 6, 8, 11, 11, 2, 7, 12

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			e(21025) = o[o[o]][o] has depth 3 so a(21025) = 3.

Cf. A007916, A052409, A052410, A109082, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.

nn=1000;
radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n===1,"o",With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Table[Max@@Length/@Position[exp[n],_],{n,200}]

A317652 Number of free pure symmetric multifunctions whose leaves are an integer partition of n.

Original entry on oeis.org

1, 1, 2, 6, 22, 93, 421, 2010, 9926, 50357, 260728, 1372436, 7321982, 39504181, 215168221, 1181540841, 6534058589, 36357935615, 203414689462, 1143589234086, 6457159029573, 36602333187792, 208214459462774, 1188252476400972, 6801133579291811, 39032172166792887

Offset: 0

Gus Wiseman, Aug 03 2018

nonn

A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.

			The a(4) = 22 free pure symmetric multifunctions:
  1[1[1[1]]]  1[1[2]]  1[3]  2[2]  4
  1[1[1][1]]  1[2[1]]  3[1]
  1[1][1[1]]  2[1[1]]
  1[1[1]][1]  1[1][2]
  1[1][1][1]  1[2][1]
  1[1[1,1]]   2[1][1]
  1[1,1[1]]   1[1,2]
  1[1][1,1]   2[1,1]
  1[1,1][1]
  1[1,1,1]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A001003, A052893, A053492, A277996, A279944, A280000.

Cf. A317653, A317654, A317655, A317656, A317658.

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]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0,{{}},If[Length[m]==1,{First[m]},Join@@Cases[Union[Table[PR[m[[s]],m[[Complement[Range[Length[m]],s]]]],{s,Take[Subsets[Range[Length[m]]],{2,-2}]}]],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h],Union[Sort/@Tuples[exprUsing/@p]]}],{p,mps[g]}]]]];
Table[Sum[Length[exprUsing[y]],{y,IntegerPartitions[n]}],{n,0,6}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[]); for(n=1, n, my(t=EulerT(v)); v=concat(v, 1 + sum(k=1, n-1, v[k]*t[n-k]))); concat([1],v)} \\ Andrew Howroyd, Aug 28 2018

Terms a(12) and beyond from Andrew Howroyd, Aug 28 2018

A317653 Number of free pure symmetric multifunctions whose leaves are a normal multiset of size n.

Original entry on oeis.org

1, 3, 34, 602, 14872, 472138, 18323359, 840503724, 44489123726, 2668985463839, 178960530393633, 13263068003965046, 1076580864432281157, 94987639225399100006, 9051397653144246683937, 926407121115738135640677, 101357200280211387377806719, 11804887470887800839909147484

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

A multiset is normal if it spans an initial interval of positive integers. A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.

			The a(3) = 34 free pure symmetric multifunctions:
1[1[1]], 1[1,1], 1[1][1],
1[2[2]], 1[2,2], 2[1[2]], 2[2[1]], 2[1,2], 1[2][2], 2[1][2], 2[2][1],
1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1],
1[2[3]], 1[3[2]], 1[2,3], 2[1[3]], 2[3[1]], 2[1,3], 3[1[2]], 3[2[1]], 3[1,2], 1[2][3], 2[1][3], 1[3][2], 3[1][2], 2[3][1], 3[2][1].

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317654, A317655, A317656, A317658.

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]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0,{},If[Length[m]==1,{First[m]},Join@@Cases[Union[Table[PR[m[[s]],m[[Complement[Range[Length[m]],s]]]],{s,Take[Subsets[Range[Length[m]]],{2,-2}]}]],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h],Union[Sort/@Tuples[exprUsing/@p]]}],{p,mps[g]}]]]];
got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
Table[Sum[Length[exprUsing[got[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,6}]

\\ here R(n,1) is A052893.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=[k]); for(n=2, n, my(t=EulerT(v)); v=concat(v, sum(k=1, n-1, v[k]*t[n-k]))); v}
seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

Terms a(8) and beyond from Andrew Howroyd, Sep 14 2018

A317994 Number of inequivalent leaf-colorings of the free pure symmetric multifunction with e-number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			Inequivalent representatives of the a(441) = 11 colorings of the expression e(441) = o[o,o][o] are the following.
  1[1,1][1]
  1[1,1][2]
  1[1,2][1]
  1[1,2][2]
  1[1,2][3]
  1[2,2][1]
  1[2,2][2]
  1[2,2][3]
  1[2,3][1]
  1[2,3][2]
  1[2,3][4]

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317765.

A317765 Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 3, 3, 4, 6, 4, 4, 5, 7, 2, 5, 5, 6, 8, 3, 6, 6, 7, 4, 9, 3, 4, 7, 7, 8, 4, 5, 10, 4, 3, 5, 8, 8, 9, 5, 6, 11, 5, 4, 6, 9, 9, 5, 10, 6, 7, 12, 6, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 7, 6, 8, 11, 11, 7, 12, 8, 9, 14, 4, 8, 7, 9, 12, 12, 3, 8

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			The a(12) = 4 subexpressions of o[o[]][] are {o, o[], o[o[]], o[o[]][]}.

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317713, A317994.

nn=1000;
radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n===1,"o",With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Table[Length[Union[Cases[exp[n],_,{0,Infinity},Heads->True]]],{n,100}]

A317655 Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 3, 8, 10, 15, 50, 35, 37, 96, 144, 160, 299, 184, 589, 840, 2483, 578, 1729, 750, 10746, 1627, 2246, 3578, 9357, 3367, 47420, 6397, 212668, 3155, 9818, 17280, 15666, 18250, 966324, 84232, 54990, 12471, 4439540, 45015

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.

			The a(6) = 8 free pure symmetric multifunctions:
  1[1[2]]
  1[2[1]]
  2[1[1]]
  1[1][2]
  1[2][1]
  2[1][1]
  1[1,2]
  2[1,1]

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317656, A317658.

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]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0,{},If[Length[m]==1,{First[m]},Join@@Cases[Union[Table[PR[m[[s]],m[[Complement[Range[Length[m]],s]]]],{s,Take[Subsets[Range[Length[m]]],{2,-2}]}]],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h],Union[Sort/@Tuples[exprUsing/@p]]}],{p,mps[g]}]]]];
got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[exprUsing[got[Reverse[primeMS[n]]]]],{n,40}]

A317656 Number of free pure symmetric multifunctions whose leaves are the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 2, 10, 1, 8, 1, 8, 2, 2, 1, 35, 1, 2, 3, 8, 1, 15, 1, 37, 2, 2, 2, 50, 1, 2, 2, 35, 1, 15, 1, 8, 8, 2, 1, 160, 1, 8, 2, 8, 1, 35, 2, 35, 2, 2, 1, 96, 1, 2, 8, 144, 2, 15, 1, 8, 2, 15, 1, 299, 1, 2, 8, 8, 2, 15, 1, 160

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.

			The a(12) = 8 free pure symmetric multifunctions are 1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1].

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317655, A317658.

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]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0,{},If[Length[m]==1,{First[m]},Join@@Cases[Union[Table[PR[m[[s]],m[[Complement[Range[Length[m]],s]]]],{s,Take[Subsets[Range[Length[m]]],{2,-2}]}]],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h],Union[Sort/@Tuples[exprUsing/@p]]}],{p,mps[g]}]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[exprUsing[primeMS[n]]],{n,100}]

cp's OEIS Frontend

A277996 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

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A317654 Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.

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A316112 Number of leaves in the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

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A317056 Depth of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

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A317652 Number of free pure symmetric multifunctions whose leaves are an integer partition of n.

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A317653 Number of free pure symmetric multifunctions whose leaves are a normal multiset of size n.

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A317994 Number of inequivalent leaf-colorings of the free pure symmetric multifunction with e-number n.

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A317765 Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

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