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A375836 Number of chains in the poset of permutations of [n].

%I A375836 #30 Jul 02 2025 04:21:48
%S A375836 1,1,3,17,165,2539,57597,1813797,75733683,4048845673,269701306809,
%T A375836 21901093760303,2129681860984785,244316156443454237,
%U A375836 32650648748310672739,5028367353617766838085,884047390780977994754809,175979907431515249448486007,39376198947363790655257792497
%N A375836 Number of chains in the poset of permutations of [n].
%H A375836 Alois P. Heinz, <a href="/A375836/b375836.txt">Table of n, a(n) for n = 0..261</a>
%F A375836 a(n) = Sum_{k=0..n} A375835(n,k).
%F A375836 a(n) = n! + Sum_{k=1..n-1} abs(Stirling1(n,k))*a(k). - _Rajesh Kumar Mohapatra_, Jul 01 2025
%F A375836 a(n) = 2 * A375838(n) - 1. - _Rajesh Kumar Mohapatra_, Jul 01 2025
%e A375836 Consider the set S = {1, 2, 3}. The a(3) = 6 + 8 + 3 = 17 in the poset of permutations of {1,2,3}:
%e A375836 |{(1)(2)(3), (1)(23), (2)(13), (3)(12), (123), (132)}| = 6;
%e A375836 |{(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123),(1)(2)(3) < (132), (1)(23) < (123), (2)(13) < (132), (3)(12) < (123)}|=8;
%e A375836 |{(1)(2)(3) < (1)(23) < (123), (1)(2)(3) < (2)(13) < (132), (1)(2)(3) < (3)(12) < (123)}| = 3.
%p A375836 a:= proc(n) option remember;
%p A375836       n!+add(abs(Stirling1(n, k))*a(k), k=1..n-1)
%p A375836     end:
%p A375836 seq(a(n), n=0..18);  # _Alois P. Heinz_, Jul 01 2025
%t A375836 b[n_, k_, t_] := b[n, k, t] = If[k < 0, 0, If[Union@{n, k} == {0}, 1, Sum[If[k == 1, 1, b[v, k - 1, 1]]*Abs[StirlingS1[n, v]], {v, k, n - t}]]];
%t A375836 a[n_] := Sum[b[n, k, 0], {k, 0, n}]; a /@ Range[0, 20]
%o A375836 (Python)
%o A375836 from math import factorial as f
%o A375836 from sympy.functions.combinatorial.numbers import stirling as s
%o A375836 from functools import cache
%o A375836 @cache
%o A375836 def a(n): return f(n) + sum(abs(s(n, k, kind=1)) * a(k) for k in range(1, n)) # _David Radcliffe_, Jul 01 2025
%Y A375836 Row sums of A375835.
%Y A375836 Cf. A048994, A331955, A330804, A331956, A331957.
%K A375836 nonn
%O A375836 0,3
%A A375836 _Rajesh Kumar Mohapatra_, Aug 31 2024