OFFSET
0,3
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.
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.
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
EXAMPLE
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}}.
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]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
S. R. Kannan and Rajesh Kumar Mohapatra, Feb 02 2020
EXTENSIONS
More terms from Michel Marcus, Feb 08 2020
STATUS
approved