A306884
Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) decreasing order.
Original entry on oeis.org
1, 1, 3, 6, 14, 28, 93, 270, 86170, 7625640881546
Offset: 0
a(0) = 1 because the empty partition () has no parts, the exponentiation operator ^ is right-associative, and 1 is the right identity of exponentiation.
a(6) = 1^1^1^1^1^1 + 2^1^1^1^1 + 2^2^1^1 + 2^2^2 + 3^1^1^1 + 3^2^1 + 3^3 + 4^1^1 + 4^2 + 5^1 + 6 = 1 + 2 + 4 + 16 + 3 + 9 + 27 + 4 + 16 + 5 + 6 = 93.
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f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
a:= n-> add(f(sort(l, `>`)), l=combinat[partition](n)):
seq(a(n), n=0..9);
A306919
Sum over all partitions of n into distinct parts of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in increasing order.
Original entry on oeis.org
1, 1, 2, 4, 5, 14, 24, 122, 318, 2417851639229258349414245, 14134776518227074636666380005943348126619871175004951664972849610340964762
Offset: 0
a(0) = 1 because the empty partition () has no parts, the exponentiation operator ^ is right-associative, and 1 is the right identity of exponentiation.
a(6) = 1^2^3 + 2^4 + 1^5 + 6 = 1 + 16 + 1 + 6 = 24.
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d:= proc(l) local i; for i to nops(l)-1 do
if l[i]=l[i+1] then return fi od; l
end:
f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
a:= n-> add(f(l), l=map(l->d(sort(l, `<`)), combinat[partition](n))):
seq(a(n), n=0..11);
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d[l_] := Module[{i}, For[i = 1, i <= Length[l]-1 , i++, If[l[[i]] == l[[i+1]], Return[]]]; l];
f[l_] := If[l == {}, 1, l[[1]]^f[Delete[l, 1]]];
a[n_] := Sum[f[l], {l, Sort /@ Select[IntegerPartitions[n], Length@# == Length @ Union@#&]}];
a /@ Range[0, 11] (* Jean-François Alcover, May 03 2020, after Maple *)
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