A375838 -id:A375838 - cp's OEIS frontend

A385521 Decimal expansion of a constant related to A375838.

1, 5, 9, 5, 8, 5, 4, 3, 3, 0, 5, 0, 0, 3, 6, 6, 2, 1, 2, 4, 7, 0, 0, 6, 5, 6, 9, 7, 4, 0, 0, 1, 6, 5, 1, 6, 9, 6, 4, 5, 0, 2, 5, 0, 5, 8, 4, 8, 3, 2, 4, 0, 6, 4, 2, 4, 7, 9, 4, 1, 8, 9, 0, 9, 3, 4, 1, 1, 9, 1, 0, 3, 8, 6, 1, 2, 7, 7, 4, 3, 8, 1, 3, 9, 3, 5, 8, 2, 4, 0, 2, 3, 5, 5, 5, 9, 9, 6, 5, 8, 7, 7, 1, 8, 3

Offset: 1

Vaclav Kotesovec, Jul 01 2025

Variant of Lengyel's constant A086053.

			1.59585433050036621247006569740016516964502505848324064247941890934119103861277...

Cf. A086053, A260932.
Cf. A086555, A246040, A375838.
Cf. A005121, A331957, A131407, A330804.

Equals lim_{n->oo} A375838(n) * 2^n * log(2)^n * n^(1-log(2)/3) / n!^2.

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

Original entry on oeis.org

1, 1, 3, 17, 165, 2539, 57597, 1813797, 75733683, 4048845673, 269701306809, 21901093760303, 2129681860984785, 244316156443454237, 32650648748310672739, 5028367353617766838085, 884047390780977994754809, 175979907431515249448486007, 39376198947363790655257792497

Offset: 0

Rajesh Kumar Mohapatra, Aug 31 2024

nonn

			Consider the set S = {1, 2, 3}. The a(3) = 6 + 8 + 3 = 17 in the poset of permutations of {1,2,3}:
|{(1)(2)(3), (1)(23), (2)(13), (3)(12), (123), (132)}| = 6;
|{(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;
|{(1)(2)(3) < (1)(23) < (123), (1)(2)(3) < (2)(13) < (132), (1)(2)(3) < (3)(12) < (123)}| = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..261

Row sums of A375835.

Cf. A048994, A331955, A330804, A331956, A331957.

a:= proc(n) option remember;
      n!+add(abs(Stirling1(n, k))*a(k), k=1..n-1)
    end:
seq(a(n), n=0..18);  # Alois P. Heinz, Jul 01 2025

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}]]];
a[n_] := Sum[b[n, k, 0], {k, 0, n}]; a /@ Range[0, 20]

from math import factorial as f
from sympy.functions.combinatorial.numbers import stirling as s
from functools import cache
@cache
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

a(n) = Sum_{k=0..n} A375835(n,k).
a(n) = n! + Sum_{k=1..n-1} abs(Stirling1(n,k))*a(k). - Rajesh Kumar Mohapatra, Jul 01 2025
a(n) = 2 * A375838(n) - 1. - Rajesh Kumar Mohapatra, Jul 01 2025

A375837 Triangle read by rows: T(n,k) is the number of rooted chains starting with the cycle (1)(2)(3)...(n) of length k of permutation poset of n letters.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 3, 0, 1, 23, 41, 18, 0, 1, 119, 455, 515, 180, 0, 1, 719, 5139, 10985, 9255, 2700, 0, 1, 5039, 62713, 222551, 334040, 225855, 56700, 0, 1, 40319, 840265, 4619447, 10899840, 12686030, 7193340, 1587600, 0, 1, 362879, 12383329, 101128653, 350413245, 620801580, 592261110, 289918440, 57153600

Offset: 0

Rajesh Kumar Mohapatra, Ranjan Kumar Dhani, and Subhashree Sahoo, Aug 31 2024

			Triangle T(n,k) begins:
  n\k | 0  1   2     3     4      5      6     7 ...
 -----+-----------------------------------------
  0   | 1;
  1   | 0, 1;
  2   | 0, 1, 1;
  3   | 0, 1, 5, 3;
  4   | 0, 1, 23, 41, 18;
  5   | 0, 1, 119, 455, 515, 180;
  6   | 0, 1, 719, 5139, 10985, 9255, 2700;
  7   | 0, 1, 5039, 62713, 222551, 334040, 225855, 56700;
  ...
The T(3, 2) = 5 chains in the poset of the permutations of {1, 2, 3} are: {(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123),(1)(2)(3) < (132)}.

Sean A. Irvine, Java program (github)

Cf. A000007 (column k=0), A057427 (column k=1), A006472 (diagonal), A375838 (row sums).

Cf. A048994, A331955, A330804, A331956, A331957, A375835, A375836.

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[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* corrected Jul 01 2025 *)

Let Stirling1(n, k) denote the unsigned Stirling numbers of the first kind (A132393).
T(0, 0) = 1, T(0, k) = 0 for k > 0 and T(n, 1) = 1 for n > 1.
T(n, k) = Sum_{i_(k-1)=k-1..n-1} (Sum_{i_(k-2)=k-2..i_(k-1) - 1} (... (Sum_{i_2=2..i_3 - 1} (Sum_{i_1=1..i_2 - 1} Stirling1(n,i_(k-1)) * Stirling1(i_(k-1),i_(k-2)) * ... * Stirling1(i_3,i_2) * Stirling1(i_2,i_1)))...)), where 2 <= k <= n.

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A385521 Decimal expansion of a constant related to A375838.

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

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A375837 Triangle read by rows: T(n,k) is the number of rooted chains starting with the cycle (1)(2)(3)...(n) of length k of permutation poset of n letters.

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