Search: a279944 -id:a279944
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A317655
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Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n.
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+0
7
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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]
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A317656
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Number of free pure symmetric multifunctions whose leaves are the integer partition with Heinz number n.
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+0
7
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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
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OFFSET
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1,6
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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].
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A317658
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Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom.
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+0
16
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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].
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EXAMPLE
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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[][][][][][][]
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A317659
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Regular triangle where T(n,k) is the number of distinct free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and k leaves.
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+0
1
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1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 43, 33, 9, 1, 0, 1, 21, 92, 118, 55, 11, 1, 0, 1, 28, 174, 341, 252, 82, 13, 1, 0, 1, 36, 302, 845, 935, 463, 115, 15, 1, 0, 1, 45, 490, 1864, 2921, 2103, 769, 153, 17, 1, 0, 1, 55, 755
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OFFSET
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1,8
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LINKS
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EXAMPLE
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The T(5,3) = 5 expressions are o[o[o]], o[o,o[]], o[][o,o], o[o][o], o[o,o][].
Triangle begins:
1
1 0
1 1 0
1 3 1 0
1 6 5 1 0
1 10 17 7 1 0
1 15 43 33 9 1 0
1 21 92 118 55 11 1 0
1 28 174 341 252 82 13 1 0
1 36 302 845 935 463 115 15 1 0
1 45 490 1864 2921 2103 769 153 17 1 0
1 55 755 3755 7981 8012 4145 1187 197 19 1 0
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MATHEMATICA
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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]}]]];
Table[Length[Select[maxUsing[n], Length[Position[#, "o"]]==k&]], {n, 12}, {k, n}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A317676
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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.
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+0
3
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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].
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A007916, A052409, A052410, A052893, A053492, A215366, A279944, A280000, A299759, A317658, A317659, A317677.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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