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A375838 Number of rooted chains starting with the cycle (1)(2)(3)...(n) in the permutation poset of [n].

%I A375838 #40 Jul 01 2025 17:14:50
%S A375838 1,1,2,9,83,1270,28799,906899,37866842,2024422837,134850653405,
%T A375838 10950546880152,1064840930492393,122158078221727119,
%U A375838 16325324374155336370,2514183676808883419043,442023695390488997377405,87989953715757624724243004,19688099473681895327628896249,4919839221134662388853128069571,1365091729320293490230304687026514
%N A375838 Number of rooted chains starting with the cycle (1)(2)(3)...(n) in the permutation poset of [n].
%H A375838 Vaclav Kotesovec, <a href="/A375838/b375838.txt">Table of n, a(n) for n = 0..260</a>
%F A375838 a(n) = Sum_{k=0..n} A375837(n,k).
%F A375838 a(n) = (A375836(n)+1)/2.
%F A375838 Conjecture: a(n) = R(n,0) where R(n,k) = (k+1) * (Sum_{i=0..n-1} R(n-1,i) + Sum_{j=0..k-1} R(n-1,j)) for 0 <= k < n, R(n,n) = 1. - _Mikhail Kurkov_, Jun 21 2025
%F A375838 a(n) ~ c * n!^2 / (2^n * log(2)^n * n^(1-log(2)/3)), where c = A385521 = 1.59585433050036621247006569740016516964502505848324064247941890934119103861277... - _Vaclav Kotesovec_, Jul 01 2025
%F A375838 a(n) = 1 + Sum_{k=1..n-1} abs(Stirling1(n,k))*a(k). - _Rajesh Kumar Mohapatra_, Jul 01 2025
%e A375838 Consider the set S = {1, 2, 3}. The a(3) = 1 + 5 + 3 = 9 in the poset of permutations of {1,2,3}:
%e A375838  |{(1)(2)(3)}| = 1;
%e A375838  |{(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123), (1)(2)(3) < (132)}|=5;
%e A375838  |{(1)(2)(3) < (1)(23) < (123), (1)(2)(3) < (2)(13)< (132), (1)(2)(3) < (3)(12) < (123)}| = 3.
%p A375838 a:= proc(n) option remember;
%p A375838       1+add(abs(Stirling1(n, k))*a(k), k=1..n-1)
%p A375838     end:
%p A375838 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 01 2025
%t A375838 T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[(n == 0 && k == 0) || k == 1, 1, Sum[If[r >= 0, Abs[StirlingS1[n, r]]*T[r, k - 1], 0], {r, k - 1, n - 1}]]]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 01 2025, after A375837 *)
%Y A375838 Row sums of A375837.
%Y A375838 Cf. A048994, A331955, A330804, A331956, A331957, A375835, A375836.
%Y A375838 Cf. A385521.
%K A375838 nonn
%O A375838 0,3
%A A375838 _Rajesh Kumar Mohapatra_, Subhashree Sahoo, and Ranjan Kumar Dhani, Sep 10 2024