OFFSET
0,9
COMMENTS
Also the number of chains of length k in unordered set partitions of {1,2,...,n} such that the first term of the chains is either {{1}, {2},...,{n}} or {{1,2,..,n}}.
Number of rooted 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.
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 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} Stirling2(n,i_(k-1)) * Stirling2(i_(k-1),i_(k-2)) * ... * Stirling2(i_3,i_2) * Stirling2(i_2,i_1)))...)), where 2 <= k <= n.
EXAMPLE
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 4 3
4 | 0 1 14 31 18
5 | 0 1 51 255 385 180
6 | 0 1 202 2066 6110 6945 2700
7 | 0 1 876 17549 90839 188510 171045 56700
...
The T(3,2) = 4 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}}.
Or,
{{1,2,3}} > {{1,2},{3}},
{{1,2,3}} > {{1,3},{2}},
{{1,2,3}} > {{1},{2,3}},
{{1,2,3}} > {{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:
T:= (n, k)-> b(n, k, 0):
seq(seq(T(n, k), k=0..n), n=0..10); # 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}]]];
T[n_, k_] := b[n, k, 0];
Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten
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)); }
T(n, k) = b(n, k, 0);
matrix(8, 8, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Feb 09 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
S. R. Kannan, Rajesh Kumar Mohapatra, Feb 02 2020
STATUS
approved