A331957 Number of rooted chains in set partitions of {1, 2, ..., n}.

1, 1, 2, 8, 64, 872, 18024, 525520, 20541392, 1036555120, 65591856032, 5085891210864, 474213645013904, 52346708185187392, 6751386193135966464, 1005991884967386086400, 171500271138273300946720, 33167303833191421470542496, 7222314392966179538774364128, 1759036134944451206655721276256

Offset: 0

Author

S. R. Kannan and Rajesh Kumar Mohapatra, Feb 02 2020

Keywords

• nonn

Comments

Also the number of chains of Stirling numbers of the second kind such that the first term of the chains is either {{1}, {2}, ..., {n}} or {{1,2,...,n}}.

Number of rooted fuzzy equivalence matrices of order n.

Examples

			The a(3) = 8 in the lattice of set partitions of {1,2,3}:
  {{1},{2},{3}},
  {{1},{2},{3}} < {{1,2},{3}},
  {{1},{2},{3}} < {{1,3},{2}},
  {{1},{2},{3}} < {{1},{2,3}},
  {{1},{2},{3}} < {{1,2,3}},
  {{1},{2},{3}} < {{1,2},{3}} < {{1,2,3}},
  {{1},{2},{3}} < {{1,3},{2}} < {{1,2,3}},
  {{1},{2},{3}} < {{1},{2,3}} < {{1,2,3}}.
Or,
  {{1,2,3}},
  {{1,2,3}} > {{1,2},{3}},
  {{1,2,3}} > {{1,3},{2}},
  {{1,2,3}} > {{1},{2,3}},
  {{1,2,3}} > {{1},{2},{3}},
  {{1,2,3}} > {{1},{2,3}} > {{1},{2},{3}},
  {{1,2,3}} > {{2},{1,3}} > {{1},{2},{3}},
  {{1,2,3}} > {{3},{1,2}} > {{1},{2},{3}}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..261
• S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
• V. Murali, Equivalent finite fuzzy sets and Stirling numbers, Inf. Sci., 174 (2005), 251-263.
• R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.

Crossrefs

Row sums of A331956.

Cf. A000110, A058692, A006472.

Cf. A008277, A048993, A328044, A330301, A330302.

Cf. A086053.

Programs

Maple

b:= proc(n, k, t) option remember; `if`(k<0 or k>n, 0, `if`(k=1 or
      {n, k}={0}, 1, add(b(v, k-1, 1)*Stirling2(n, v), v=k..n-t)))
    end:
a:= n-> add(b(n, k, 0), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 09 2020

Mathematica

b[n_, k_, t_] := b[n, k, t] = If[k < 0 || k > n, 0, If[k == 1 || Union@{n, k} =={0}, 1, Sum[b[v, k - 1, 1]*StirlingS2[n, v], {v, k, n - t}]]];
a[n_] := Sum[b[n, k, 0], {k, 0, n}];
a /@ Range[0, 30]

PARI

b(n, k, t) = {if (k < 0, return(0)); if ((n==0) && (k==0), return (1)); if ((k==1) && (n>0), return(1)); sum(v = k, n - t, if (k==1, 1, b(v, k-1, 1))*stirling(n, v, 2));}
a(n) = sum(k=0, n, b(n, k, 0); ); \\ Michel Marcus, Feb 09 2020

Python

from sympy.functions.combinatorial.numbers import stirling as s
from functools import cache
@cache
def a(n): return 1 + sum(s(n, k) * a(k) for k in range(1, n)) # David Radcliffe, Jul 01 2025

Formula

a(n) = Sum_{k=0..n} A331956(n,k).

Conjecture from Mikhail Kurkov, Jun 25 2025: (Start)

a(n) = R(n,0) where

R(0,0) = 1,

R(n,k) = (k+1) * Sum_{j=k..n-1} R(n-1,j) for 0 <= k < n,

R(n,n) = Sum_{j=0..n-1} R(n,j). (End)

a(n) ~ A086053 * n!^2 / (2^(n-1) * log(2)^n * n^(1 + log(2)/3)). - Vaclav Kotesovec, Jul 01 2025

a(n) = 1 + Sum_{k=1..n-1} Stirling2(n,k)*a(k). - Rajesh Kumar Mohapatra, Jul 01 2025

Extensions

More terms from Michel Marcus, Feb 08 2020