A052893 -id:A052893 - cp's OEIS frontend

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

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}]

A300626 Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions.

Original entry on oeis.org

1, 1, 3, 11, 43, 187, 872, 4375, 23258, 130485, 767348, 4710715, 30070205, 198983975, 1361361925, 9607908808, 69812787049, 521377973359, 3996036977270, 31389624598631, 252408597286705, 2075472033455894, 17434190966525003, 149476993511444023, 1307022313790487959

Offset: 0

Gus Wiseman, Aug 17 2018

nonn

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

Also the number of inequivalent colorings of orderless Mathematica expressions with n positions.

			Inequivalent representatives of the a(3) = 11 colorings:
  1[1,1]  1[2,2]  1[1,2]  1[2,3]
  1[1[]]  1[2[]]
  1[][1]  1[][2]
  1[1][]  1[2][]
  1[][][]

Row sums of A304485.

Cf. A000612, A007716, A052893, A053492, A277996, A279944, A280000, A317652, A317655, A317656, A317676.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020

A317882 Number of free pure achiral multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 1, 2, 5, 12, 31, 79, 211, 564, 1543, 4259, 11899, 33526, 95272, 272544, 784598, 2270888, 6604900, 19293793, 56581857, 166523462, 491674696, 1455996925, 4323328548, 12869353254, 38396655023, 114803257039, 343932660450, 1032266513328, 3103532577722

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A free pure achiral multifunction (with empty expressions allowed) (AME) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g, ..., g] where h and g are AMEs. The number of positions in an AME is the number of brackets [...] plus the number of o's.

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

			The a(5) = 12 AMEs:
  o[o[o]]
  o[o][o]
  o[o[][]]
  o[o,o,o]
  o[][o[]]
  o[][o,o]
  o[][][o]
  o[o[]][]
  o[o,o][]
  o[][o][]
  o[o][][]
  o[][][][]

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

Cf. A001003, A002033, A003238, A052893, A053492, A214577, A277996, A317853, A317875.

Cf. A317883, A317884, A317885.

a[n_]:=If[n==1,1,Sum[a[k]*If[k==n-1,1,Sum[a[d],{d,Divisors[n-k-1]}]],{k,n-1}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*(1 + sum(k=1, n-2, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=v[n-1] + sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = a(n - 1) + Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018

A317883 Number of free pure achiral multifunctions with one atom and n positions.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 10, 17, 37, 70, 150, 299, 634, 1311, 2786, 5879, 12584, 26904, 58005, 125242, 271819, 591297, 1290976, 2825170, 6199964, 13635749, 30057649, 66386206, 146903289, 325637240, 723024160, 1607805207, 3580476340, 7984266625, 17827226469

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A free pure achiral multifunction (PAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g] where h and g are PAMs. The number of positions in a PAM is the number of brackets [...] plus the number of o's.

			The a(7) = 10 PAMs:
  o[o[o[o]]]
  o[o[o][o]]
  o[o][o[o]]
  o[o[o]][o]
  o[o][o][o]
  o[o[o,o,o]]
  o[o][o,o,o]
  o[o,o][o,o]
  o[o,o,o][o]
  o[o,o,o,o,o]

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

Cf. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.

Cf. A317882, A317884, A317885.

a[n_]:=If[n==1,1,Sum[a[k]*Sum[a[d],{d,Divisors[n-k-1]}],{k,n-2}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*sum(k=1, n-2, subst(p + O(x^(n\k+1)), x, x^k) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).

G.f. A(x) satisfies: A(x) = x * (1 + A(x) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, May 03 2019

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018

A317884 Number of series-reduced achiral free pure multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 14, 26, 47, 87, 160, 295, 540, 997, 1832, 3369, 6197, 11406, 20975, 38594, 70991, 130610, 240275, 442043, 813184, 1496053, 2752251, 5063319, 9314879, 17136632, 31526032, 57998423, 106699160, 196294065, 361120800, 664352454, 1222204958

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A series-reduced achiral expression (SRAE) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty but not unitary expression of the form h[g, ..., g], where h and g are SRAEs. The number of positions in an SRAE is the number of brackets [...] plus the number of o's.

Also the number of series-reduced achiral Mathematica expressions with one atom and n positions.

			The a(6) = 8 SRAEs:
  o[o,o,o,o]
  o[o[],o[]]
  o[][o,o,o]
  o[][][o,o]
  o[o,o,o][]
  o[][o,o][]
  o[o,o][][]
  o[][][][][]

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.

Cf. A317882, A317883, A317885.

a:= proc(n) option remember; `if`(n=1, 1, a(n-1)+add(a(j)*add(
      a(d), d=numtheory[divisors](n-j-1) minus {n-j-1}), j=1..n-1))
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, Sep 05 2018

allAchExprSR[n_] := If[n == 1, {"o"}, Join @@ Cases[Table[PR[k, n - k - 1], {k, n - 1}], PR[h_, g_] :> Join @@ Table[Apply @@@ Tuples[{allAchExprSR[h], Select[Tuples[allAchExprSR /@ p], SameQ @@ # &]}], {p, If[g == 0, {{}}, Join @@ Permutations /@ Rest[IntegerPartitions[g]]]}]]]; Table[Length[allAchExprSR[n]], {n, 12}]
(* Second program: *)
a[n_] := a[n] = If[n == 1, 1, a[n-1] + Sum[a[j]*DivisorSum[
     n-j-1, If[# < n-j-1, a[#], 0]&], {j, 1, n-2}]];
Array[a, 45] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*(1 + sum(k=2, n-2, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=v[n-1] + sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, if(dAndrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = a(n-1) + Sum_{0 < k < n-1} a(k) * Sum_{d|(n-k-1), d < n-k-1} a(d).

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018

A317885 Number of series-reduced free pure achiral multifunctions with one atom and n positions.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 9, 14, 21, 32, 45, 69, 103, 153, 224, 338, 500, 746, 1107, 1645, 2447, 3652, 5413, 8052, 11993, 17834, 26500, 39447, 58655, 87240, 129772, 193001, 287034, 427014, 635048, 944501, 1404910, 2089633, 3107864, 4622670, 6875533

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A series-reduced free pure achiral multifunction (SRAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty and non-unitary expression of the form h[g, ..., g] where h and g are SRAMs. The number of positions in a SRAM is the number of brackets [...] plus the number of o's.

			The a(10) = 7 SRAMs:
  o[o[o,o],o[o,o]]
  o[o,o][o,o][o,o]
  o[o,o][o,o,o,o,o]
  o[o,o,o][o,o,o,o]
  o[o,o,o,o][o,o,o]
  o[o,o,o,o,o][o,o]
  o[o,o,o,o,o,o,o,o]

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A317853, A317875.

Cf. A317882, A317883, A317884.

a[n_]:=If[n==1,1,Sum[a[k]*Sum[a[d],{d,Most[Divisors[n-k-1]]}],{k,n-2}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*sum(k=2, n-2, subst(p + O(x^(n\k+1)), x, x^k)) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, if(dAndrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1), d < n - k - 1} a(d).

Terms a(17) and beyond from Andrew Howroyd, Aug 19 2018

A317853 a(1) = 1; a(n > 1) = Sum_{0 < k < n} (-1)^(n - k - 1) a(n - k) Sum_{d|k} a(d).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 6, 11, 14, 23, 26, 51, 70, 114, 147, 237, 314, 516, 715, 1118, 1549, 2353, 3252, 5011, 7235, 10724, 15142, 22504, 32506, 47770, 69173, 100980, 146657, 212504, 308563, 448256, 658037, 946166, 1373739, 1988283, 2919185, 4197886, 6118850

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317875.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.

a[n_]:=a[n]=If[n==1,1,Sum[(-1)^(n-k-1)*a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,50]

A317676 Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 16, 7, 10, 12, 13, 21, 25, 27, 32, 36, 64, 81, 128, 256, 11, 14, 17, 18, 28, 33, 35, 41, 45, 49, 75, 93, 100, 125, 144, 145, 169, 216, 243, 279, 441, 512, 625, 729, 1024, 1296, 2048, 2187, 4096, 6561, 8192, 16384, 65536, 524288, 8388608, 9007199254740992

Offset: 1

Gus Wiseman, Aug 03 2018

Given a positive integer n we construct a unique free pure symmetric multifunction e(n) 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)].

Every free pure symmetric multifunction (with empty expressions allowed) f with one atom and n positions has a unique e-number n such that e(n) = f, and vice versa, so this sequence is a permutation of the positive integers.

			Triangle begins:
  1
  2
  3   4
  5   6   8   9  16
  7  10  12  13  21  25  27  32  36  64  81 128 256
Corresponding triangle of free pure symmetric multifunctions (with empty expressions allowed) begins:
  o,
  o[],
  o[][], o[o],
  o[][][], o[o][], o[o[]], o[][o], o[o,o].

Mathematica Reference, Orderless

Row lengths are A277996.

Cf. A007916, A052409, A052410, A052893, A053492, A215366, A279944, A280000, A299759, A317658, A317659, A317677.

maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];
radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]==1];
Clear[rad];rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
ungo[x_?AtomQ]:=1;ungo[h_[g___]]:=rad[ungo[h]]^(Times@@Prime/@ungo/@{g});
Table[Sort[ungo/@maxUsing[n]],{n,5}]

A052891 Number of objects generated by the Combstruct grammar defined in the Maple program. See the link for the grammar specification.

Original entry on oeis.org

0, 1, 2, 5, 16, 56, 217, 876, 3686, 15903, 70103, 314042, 1426076, 6548060, 30352695, 141837086, 667469159, 3160370217, 15045244375, 71970393570, 345766441537, 1667629158127, 8071308125136, 39190243658297, 190845259909328, 931856232714004, 4561292365652751

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 867
Maplesoft, Combstruct grammars.

Cf. A052893.

spec := [S, {C=Prod(Z,B), S=Set(C,1 <= card), B=Sequence(S)}, unlabeled]:
seq(combstruct[count](spec,size=n), n=0..20);

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[0]); for(n=1, n, v=concat([0],EulerT(Vec(1/(1-Ser(v)))))); v} \\ Andrew Howroyd, Aug 09 2020

G.f.: 1 - 1/g(x) where g(x) is the g.f. of A052893. - Andrew Howroyd, Aug 09 2020

Terms a(21) and beyond from Andrew Howroyd, Aug 09 2020

cp's OEIS Frontend

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

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

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A300626 Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions.

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PARI

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A317882 Number of free pure achiral multifunctions (with empty expressions allowed) with one atom and n positions.

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A317883 Number of free pure achiral multifunctions with one atom and n positions.

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Mathematica

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A317884 Number of series-reduced achiral free pure multifunctions (with empty expressions allowed) with one atom and n positions.

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Maple

Mathematica

PARI

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A317885 Number of series-reduced free pure achiral multifunctions with one atom and n positions.

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