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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.

%I A306919 #17 Feb 16 2025 08:33:55
%S A306919 1,1,2,4,5,14,24,122,318,2417851639229258349414245,
%T A306919 14134776518227074636666380005943348126619871175004951664972849610340964762
%N 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.
%H A306919 Alois P. Heinz, <a href="/A306919/b306919.txt">Table of n, a(n) for n = 0..11</a>
%H A306919 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A306919 Wikipedia, <a href="https://en.wikipedia.org/wiki/Exponentiation">Exponentiation</a>
%H A306919 Wikipedia, <a href="https://en.wikipedia.org/wiki/Identity_element">Identity element</a>
%H A306919 Wikipedia, <a href="https://en.wikipedia.org/wiki/Operator_associativity">Operator associativity</a>
%H A306919 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A306919 a(0) = 1 because the empty partition () has no parts, the exponentiation operator ^ is right-associative, and 1 is the right identity of exponentiation.
%e A306919 a(6) = 1^2^3 + 2^4 + 1^5 + 6 = 1 + 16 + 1 + 6 = 24.
%p A306919 d:= proc(l) local i; for i to nops(l)-1 do
%p A306919        if l[i]=l[i+1] then return fi od; l
%p A306919     end:
%p A306919 f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
%p A306919 a:= n-> add(f(l), l=map(l->d(sort(l, `<`)), combinat[partition](n))):
%p A306919 seq(a(n), n=0..11);
%t A306919 d[l_] := Module[{i}, For[i = 1, i <= Length[l]-1 , i++, If[l[[i]] == l[[i+1]], Return[]]]; l];
%t A306919 f[l_] := If[l == {}, 1, l[[1]]^f[Delete[l, 1]]];
%t A306919 a[n_] := Sum[f[l], {l, Sort /@ Select[IntegerPartitions[n], Length@# == Length @ Union@#&]}];
%t A306919 a /@ Range[0, 11] (* _Jean-François Alcover_, May 03 2020, after Maple *)
%Y A306919 Cf. A022629, A066189, A306895, A306918.
%K A306919 nonn
%O A306919 0,3
%A A306919 _Alois P. Heinz_, Mar 16 2019