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Search: a279944 -id:a279944
Displaying 1-10 of 15 results found. page 1 2
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A277996 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions. +10
33
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Also the number of distinct orderless Mathematica expressions with one atom and n positions.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
Mathematica Reference, Orderless.
FORMULA
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)
EXAMPLE
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[][][][]
MATHEMATICA
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]
PROG
(PARI) 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
CROSSREFS
Cf. A000108, A001003, A005043, A052893, A279944, A280000, A317658.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 24 2016
STATUS
approved
A280000 Number of free pure symmetric multifunctions in one symbol with n positions. +10
28
1, 0, 1, 1, 3, 5, 12, 25, 57, 128, 296, 688, 1618, 3839, 9170, 22065, 53370, 129807, 317080, 777887, 1915247, 4731932, 11726476, 29143123, 72614115, 181363151, 453975928, 1138697689, 2861607677, 7204169689 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
A free pure symmetric multifunction (PSM) in one symbol x is either (case 1) the symbol x, or (case 2) an expression of the form h[g_1,...,g_k] where h is a PSM in x, each of the g_i for i=1..(k>0) is a PSM in x, and for i < j we have g_i <= g_j under a canonical total ordering such as the Mathematica ordering. The number of positions in a PSM is the number of brackets [...] plus the number of x's.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
EXAMPLE
Sequence of free pure symmetric multifunctions (second column) together with their numbers of positions (first column) and j-numbers (third column, see A279944 for details) begins:
1 x 1
3 x[x] 2
4 x[x,x] 8
5 x[x][x] 3
5 x[x[x]] 4
5 x[x,x,x] 128
6 x[x,x][x] 12
6 x[x][x,x] 27
6 x[x,x[x]] 32
6 x[x,x,x,x] 32768
6 x[x[x,x]] 262144
7 x[x][x][x] 5
7 x[x[x]][x] 6
7 x[x][x[x]] 9
7 x[x[x][x]] 16
7 x[x[x[x]]] 64
7 x[x,x,x][x] 145
7 x[x,x][x,x] 1728
7 x[x,x,x[x]] 2048
7 x[x][x,x,x] 2187
7 x[x,x,x,x,x] 2147483648
7 x[x,x[x,x]] 137438953472
7 x[x[x,x,x]] 1378913...3030144
MATHEMATICA
multing[t_, n_]:=Array[(t+#-1)/#&, n, 1, Times];
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-2}]];
Array[a, 15]
PROG
(PARI) 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, sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018
CROSSREFS
Cf. A005043 (non-symmetric case), A279944.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 24 2016
STATUS
approved
A052893 Number of objects generated by the Combstruct grammar defined in the Maple program. See the link for the grammar specification. +10
23
1, 1, 3, 10, 37, 144, 589, 2483, 10746, 47420, 212668, 966324, 4439540, 20587286, 96237484, 453012296, 2145478716, 10215922013, 48877938369, 234862013473, 1132902329028, 5483947191651, 26630419098206, 129696204701807, 633339363924611, 3100369991303297 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Number of free pure symmetric multifunctions with n + 1 unlabeled leaves. A free pure symmetric multifunction f in PSM is either (case 1) f = the leaf symbol "o", or (case 2) f = an expression of the form h[g_1, ..., g_k] where k > 0, h is in PSM, each of the g_i for i = 1, ..., k is in PSM, and for i < j we have g_i <= g_j under a canonical total ordering of PSM, such as the Mathematica ordering of expressions. - Gus Wiseman, Aug 02 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 869
Maplesoft, Combstruct grammars.
Mathematica Reference, Orderless.
FORMULA
G.f.: 1/(1 - g(x)) where g(x) is the g.f. of A052891. - Andrew Howroyd, Aug 09 2020
EXAMPLE
From Gus Wiseman, Aug 02 2018: (Start)
The a(3) = 10 free pure symmetric multifunctions with 4 unlabeled leaves:
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]
(End)
MAPLE
spec := [S, {C = Set(B, 1 <= card), B=Prod(Z, S), S=Sequence(C)}, unlabeled]:
seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
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]}], {k, 1, n-1}]];
Array[a, 30] (* Gus Wiseman, Aug 02 2018 *)
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=1, n, v=Vec(1/(1-x*Ser(EulerT(v))))); v} \\ Andrew Howroyd, Aug 09 2020
CROSSREFS
Cf. A001003, A052891, A277996, A279944, A280000.
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from Gus Wiseman, Aug 02 2018
STATUS
approved
A317658 Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom. +10
16
1, 2, 3, 3, 4, 4, 5, 4, 4, 5, 6, 5, 5, 6, 7, 4, 6, 6, 7, 8, 5, 7, 7, 8, 5, 9, 5, 6, 8, 8, 9, 5, 6, 10, 6, 5, 7, 9, 9, 10, 6, 7, 11, 7, 6, 8, 10, 10, 6, 11, 7, 8, 12, 8, 7, 9, 11, 11, 7, 12, 8, 9, 13, 5, 9, 8, 10, 12, 12, 8, 13, 9, 10, 14, 6, 10, 9, 11, 13, 13 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Given a positive integer n > 1 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)].
Also the number of positions in the orderless Mathematica expression with e-number n.
LINKS
Table of n, a(n) for n=1..80.
Mathematica Reference, Orderless
FORMULA
a(rad(x)^(prime(y_1) * ... * prime(y_k)) = a(x) + a(y_1) + ... + a(y_k).
e(2^(2^n)) = o[o,...,o].
e(2^prime(2^prime(2^...))) = o[o[...o[o]]].
e(rad(rad(rad(...)^2)^2)^2) = o[o][o]...[o].
EXAMPLE
The first twenty Mathematica expressions:
1: o
2: o[]
3: o[][]
4: o[o]
5: o[][][]
6: o[o][]
7: o[][][][]
8: o[o[]]
9: o[][o]
10: o[o][][]
11: o[][][][][]
12: o[o[]][]
13: o[][o][]
14: o[o][][][]
15: o[][][][][][]
16: o[o,o]
17: o[o[]][][]
18: o[][o][][]
19: o[o][][][][]
20: o[][][][][][][]
MATHEMATICA
nn=100;
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, x, 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[exp[n], {n, 1, nn}]
CROSSREFS
First differs from A277615 at a(128) = 5, A277615(128) = 6.
Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
Cf. A317652, A317653, A317654, A317655, A317656.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
STATUS
approved
A317654 Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n. +10
10
1, 3, 26, 375, 6696, 159837, 4389226, 144915350, 5377002075, 227624621051, 10632808475596, 550932945236121, 31062550998284221, 1907051034025848314, 126052420069459211076, 8956882232940915920404, 679298518935625486287703, 54868537321267493152151502, 4696952405203792017289469056 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..19.
EXAMPLE
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].
MATHEMATICA
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}]
PROG
(PARI) \\ 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
CROSSREFS
Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
Cf. A317652, A317653, A317655, A317656, A317658.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, Jan 01 2021
STATUS
approved
A317652 Number of free pure symmetric multifunctions whose leaves are an integer partition of n. +10
9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
EXAMPLE
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]
MATHEMATICA
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}]
PROG
(PARI) 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
CROSSREFS
Cf. A001003, A052893, A053492, A277996, A279944, A280000.
Cf. A317653, A317654, A317655, A317656, A317658.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Aug 28 2018
STATUS
approved
A317653 Number of free pure symmetric multifunctions whose leaves are a normal multiset of size n. +10
9
1, 3, 34, 602, 14872, 472138, 18323359, 840503724, 44489123726, 2668985463839, 178960530393633, 13263068003965046, 1076580864432281157, 94987639225399100006, 9051397653144246683937, 926407121115738135640677, 101357200280211387377806719, 11804887470887800839909147484 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
EXAMPLE
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].
MATHEMATICA
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}]
PROG
(PARI) \\ 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
CROSSREFS
Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
Cf. A317652, A317654, A317655, A317656, A317658.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, Sep 14 2018
STATUS
approved
A317655 Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n. +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..42.
EXAMPLE
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]
MATHEMATICA
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}]
CROSSREFS
Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
Cf. A317652, A317653, A317654, A317656, A317658.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
STATUS
approved
A317656 Number of free pure symmetric multifunctions whose leaves are the integer partition with Heinz number n. +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..80.
EXAMPLE
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].
MATHEMATICA
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}]
CROSSREFS
Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
Cf. A317652, A317653, A317654, A317655, A317658.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
STATUS
approved
A300626 Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..24.
EXAMPLE
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[][][]
PROG
(PARI) \\ 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
CROSSREFS
Row sums of A304485.
Cf. A000612, A007716, A052893, A053492, A277996, A279944, A280000, A317652, A317655, A317656, A317676.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 17 2018
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020
STATUS
approved
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Last modified August 30 00:57 EDT 2024. Contains 375520 sequences. (Running on oeis4.)