A331956 Triangle T(n,k) read by rows: number of rooted chains of length k in set partitions of n labeled points.

1, 0, 1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 14, 31, 18, 0, 1, 51, 255, 385, 180, 0, 1, 202, 2066, 6110, 6945, 2700, 0, 1, 876, 17549, 90839, 188510, 171045, 56700, 0, 1, 4139, 159615, 1364307, 4603620, 7314650, 5507460, 1587600

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

Cf. A000007 (column k=0), A057427 (column k=1), A058692 (column k=2), A006472 (diagonal), A331957 (row sums).

Cf. A000110, A008277, A048993, A328044, A330301, A330302.

Sequence in context: A346366 A325011 A294188 * A325019 A152151 A327125

Adjacent sequences: A331953 A331954 A331955 * A331957 A331958 A331959

Keyword

nonn,tabl

Author

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

Status

approved