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%I #35 Aug 16 2023 14:44:17
%S 1,1,2,4,12,42,216,1200,8664,66384,612264,5910024,66723384,776642664,
%T 10311400344,141065450904,2153769250584,33743736435864,
%U 583781959921944,10308436641381144,198863818304824344,3914117125411211544,83301822014343774744,1805447764831655109144
%N Number of permutations of [n] whose cycle lengths are nondecreasing when cycles are ordered by their minima and these minima are {1..k} (for some k <= n).
%C The original name was: Row sums of A179972 and also of A179974.
%H Alois P. Heinz, <a href="/A179973/b179973.txt">Table of n, a(n) for n = 0..450</a>
%F From _Alois P. Heinz_, Jul 09 2023: (Start)
%F a(n) = Sum_{lambda in partitions(n)} (n - |lambda|)!.
%F Limit_{n->oo} A004086(a(n))/10^A055642(a(n)) = A364128. (End)
%e a(4) = 12 = 6 + 2 + 2 + 1 + 1: (1234), (1243), (1324), (1342), (1423), (1432),
%e (13)(24), (14)(23), (1)(234), (1)(243), (1)(2)(34), (1)(2)(3)(4).
%p a:= n-> add((n-nops(p))!, p=combinat[partition](n)):
%p seq(a(n), n=0..24); # _Alois P. Heinz_, Jul 09 2023
%p # second Maple program:
%p b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p (p-n)!, b(n, i-1, p)+b(n-i, min(n-i, i), p-1))
%p end:
%p a:= n-> b(n$3):
%p seq(a(n), n=0..24); # _Alois P. Heinz_, Jul 09 2023
%t b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p - n)!, b[n, i - 1, p] + b[n - i, Min[n - i, i], p - 1]];
%t a[n_] := b[n, n, n];
%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Aug 16 2023, after _Alois P. Heinz_ *)
%Y Cf. A000041, A000142, A000522, A004086, A053529, A179972, A179974, A327711, A362362, A364128.
%K nonn
%O 0,3
%A _Alford Arnold_, Aug 05 2010
%E Edited by _R. J. Mathar_, May 17 2016
%E a(0), a(9)-a(23) and new name from _Alois P. Heinz_, Jul 09 2023
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