OFFSET
0,5
COMMENTS
Also the number of chains of equivalence relations of length k on a set of n-points.
Number of chains of length k in Stirling numbers of the second kind.
Number of chains of length k in the unordered partition of {1,2,...,n}.
Number of k-level fuzzy equivalence matrices of order n.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
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.
V. Murali, Combinatorics of counting finite fuzzy subsets, Fuzzy Sets Syst., 157(17)(2006), 2403-2411.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.
FORMULA
T(0, 0) = 1, T(0, k) = 0 for k > 0.
T(n, k) = Sum_{i_k=k..n} (Sum_{i_(k-1)=k-1..i_k - 1} (... (Sum_{i_2=2..i_3 - 1} (Sum_{i_1=1..i_2 - 1} Stirling2(n,i_k) * Stirling2(i_k,i_(k-1)) * ... * Stirling2(i_3,i_2) * Stirling2(i_2,i_1)))...)), where 1 <= k <= n.
EXAMPLE
The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7...
0 1
1 0 1
2 0 2 1
3 0 5 7 3
4 0 15 45 49 18
5 0 52 306 640 565 180
6 0 203 2268 8176 13055 9645 2700
7 0 877 18425 108388 279349 359555 227745 56700
...
The T(3,2) = 7 in the lattice of set partitions of {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},
{{1,3},{2}} < {{1,2,3}},
{{1},{2,3}} < {{1,2,3}}.
MAPLE
b:= proc(n, k, t) option remember; `if`(k<0, 0, `if`({n, k}={0}, 1,
add(`if`(k=1, 1, b(v, k-1, 1))*Stirling2(n, v), v=k..n-t)))
end:
T:= (n, k)-> b(n, k, 0):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Feb 07 2020
MATHEMATICA
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]]*StirlingS2[n, v], {v, k, n - t}]]];
T[n_, k_] := b[n, k, 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 08 2020, after Alois P. Heinz *)
PROG
(PARI)
b(n, k, t) = {if (k < 0, return(0)); if ((n==0) && (k==0), return (1)); sum(v = k, n - t, if (k==1, 1, b(v, k-1, 1))*stirling(n, v, 2)); }
T(n, k) = b(n, k, 0);
matrix(8, 8, n, k, T(n-1, k-1) \\ to see the triangle \\ Michel Marcus, Feb 08 2020
CROSSREFS
T(2n,n) gives A332244.
KEYWORD
nonn,tabl
AUTHOR
S. R. Kannan, Rajesh Kumar Mohapatra, Feb 02 2020
STATUS
approved