A331955 - OEIS

A331955

Triangle T(n,k) of number of chains of length k in partitions of an n-set ordered by refinement.

%I #51 Jan 14 2024 12:10:44

%S 1,0,1,0,2,1,0,5,7,3,0,15,45,49,18,0,52,306,640,565,180,0,203,2268,

%T 8176,13055,9645,2700,0,877,18425,108388,279349,359555,227745,56700,0,

%U 4140,163754,1523922,5967927,11918270,12822110,7095060,1587600

%N Triangle T(n,k) of number of chains of length k in partitions of an n-set ordered by refinement.

%C Also the number of chains of equivalence relations of length k on a set of n-points.

%C Number of chains of length k in Stirling numbers of the second kind.

%C Number of chains of length k in the unordered partition of {1,2,...,n}.

%C Number of k-level fuzzy equivalence matrices of order n.

%H Alois P. Heinz, <a href="/A331955/b331955.txt">Rows n = 0..140, flattened</a>

%H S. R. Kannan and Rajesh Kumar Mohapatra, <a href="https://arxiv.org/abs/1909.13678">Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques</a>, arXiv preprint arXiv:1909.13678 [math.GM], 2019.

%H V. Murali, <a href="https://doi.org/10.1016/j.ins.2004.08.008">Equivalent finite fuzzy sets and Stirling numbers</a>, Inf. Sci., 174 (2005), 251-263.

%H V. Murali, <a href="https://doi.org/10.1016/j.fss.2006.03.005">Combinatorics of counting finite fuzzy subsets</a>, Fuzzy Sets Syst., 157(17)(2006), 2403-2411.

%H R. B. Nelsen and H. Schmidt, Jr., <a href="http://www.jstor.org/stable/2690450">Chains in power sets</a>, Math. Mag., 64 (1) (1991), 23-31.

%F T(0, 0) = 1, T(0, k) = 0 for k > 0.

%F 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.

%e The triangle T(n,k) begins:

%e n\k 0 1 2 3 4 5 6 7...

%e 0 1

%e 1 0 1

%e 2 0 2 1

%e 3 0 5 7 3

%e 4 0 15 45 49 18

%e 5 0 52 306 640 565 180

%e 6 0 203 2268 8176 13055 9645 2700

%e 7 0 877 18425 108388 279349 359555 227745 56700

%e ...

%e The T(3,2) = 7 in the lattice of set partitions of {1,2,3}:

%e {{1},{2},{3}} < {{1,2},{3}},

%e {{1},{2},{3}} < {{1,3},{2}},

%e {{1},{2},{3}} < {{1},{2,3}},

%e {{1},{2},{3}} < {{1,2,3}},

%e {{1,2},{3}} < {{1,2,3},

%e {{1,3},{2}} < {{1,2,3}},

%e {{1},{2,3}} < {{1,2,3}}.

%p b:= proc(n, k, t) option remember; `if`(k<0, 0, `if`({n, k}={0}, 1,

%p add(`if`(k=1, 1, b(v, k-1, 1))*Stirling2(n, v), v=k..n-t)))

%p end:

%p T:= (n, k)-> b(n, k, 0):

%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Feb 07 2020

%t 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 T[n_, k_] := b[n, k, 0];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 08 2020, after _Alois P. Heinz_ *)

%o (PARI)

%o 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));}

%o T(n, k) = b(n, k, 0);

%o matrix(8, 8, n, k, T(n-1, k-1) \\ to see the triangle \\ _Michel Marcus_, Feb 08 2020

%Y Cf. A000007 (column k=0), A000110 (column k=1), A006472 (diagonal), A330804 (row sums).

%Y T(2n,n) gives A332244.

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

%K nonn,tabl

%O 0,5

%A S. R. Kannan, _Rajesh Kumar Mohapatra_, Feb 02 2020