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A375837 Triangle read by rows: T(n,k) is the number of rooted chains starting with the cycle (1)(2)(3)...(n) of length k of permutation poset of n letters.

%I A375837 #25 Jul 08 2025 23:31:58
%S A375837 1,0,1,0,1,1,0,1,5,3,0,1,23,41,18,0,1,119,455,515,180,0,1,719,5139,
%T A375837 10985,9255,2700,0,1,5039,62713,222551,334040,225855,56700,0,1,40319,
%U A375837 840265,4619447,10899840,12686030,7193340,1587600,0,1,362879,12383329,101128653,350413245,620801580,592261110,289918440,57153600
%N A375837 Triangle read by rows: T(n,k) is the number of rooted chains starting with the cycle (1)(2)(3)...(n) of length k of permutation poset of n letters.
%H A375837 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a375/A375837.java">Java program</a> (github)
%F A375837 Let Stirling1(n, k) denote the unsigned Stirling numbers of the first kind (A132393).
%F A375837 T(0, 0) = 1, T(0, k) = 0 for k > 0 and T(n, 1) = 1 for n > 1.
%F A375837 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} Stirling1(n,i_(k-1)) * Stirling1(i_(k-1),i_(k-2)) * ... * Stirling1(i_3,i_2) * Stirling1(i_2,i_1)))...)), where 2 <= k <= n.
%e A375837 Triangle T(n,k) begins:
%e A375837   n\k | 0  1   2     3     4      5      6     7 ...
%e A375837  -----+-----------------------------------------
%e A375837   0   | 1;
%e A375837   1   | 0, 1;
%e A375837   2   | 0, 1, 1;
%e A375837   3   | 0, 1, 5, 3;
%e A375837   4   | 0, 1, 23, 41, 18;
%e A375837   5   | 0, 1, 119, 455, 515, 180;
%e A375837   6   | 0, 1, 719, 5139, 10985, 9255, 2700;
%e A375837   7   | 0, 1, 5039, 62713, 222551, 334040, 225855, 56700;
%e A375837   ...
%e A375837 The T(3, 2) = 5 chains in the poset of the permutations of {1, 2, 3} are: {(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123),(1)(2)(3) < (132)}.
%t A375837 T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[(n == 0 && k == 0) || k == 1, 1, Sum[If[r >= 0, Abs[StirlingS1[n, r]]*T[r, k - 1], 0], {r, k - 1, n - 1}]]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* corrected Jul 01 2025 *)
%Y A375837 Cf. A000007 (column k=0), A057427 (column k=1), A006472 (diagonal), A375838 (row sums).
%Y A375837 Cf. A048994, A331955, A330804, A331956, A331957, A375835, A375836.
%K A375837 nonn,tabl
%O A375837 0,9
%A A375837 _Rajesh Kumar Mohapatra_, Ranjan Kumar Dhani, and Subhashree Sahoo, Aug 31 2024