This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A375835 #16 Sep 07 2024 08:29:46
%S A375835 1,0,1,0,2,1,0,6,8,3,0,24,64,59,18,0,120,574,970,695,180,0,720,5858,
%T A375835 16124,20240,11955,2700,0,5040,67752,285264,556591,559895,282555,
%U A375835 56700,0,40320,880584,5459712,15519287,23585870,19879370,8780940,1587600,0,362880,12746208,113511982,451541898,971214825,1213062690,882179550,347072040,57153600
%N A375835 Triangle read by rows: T(n, k) is the number of chains of length k in the poset of permutations of an n-set.
%F A375835 Let Stirling1(n, k) denote the unsigned Stirling numbers of the first kind (A132393).
%F A375835 T(0, 0) = 1, T(0, k) = 0 for k > 0.
%F A375835 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} Stirling1(n, i_k) * Stirling1(i_k, i_(k-1)) * ... * Stirling1(i_3, i_2) * Stirling1(i_2, i_1)))...)), where 1 <= k <= n.
%e A375835 The triangle T(n,k) begins:
%e A375835 n\k 0 1 2 3 4 5 6 7 ...
%e A375835 0 1
%e A375835 1 0 1
%e A375835 2 0 2 1
%e A375835 3 0 6 8 3
%e A375835 4 0 24 64 59 18
%e A375835 5 0 120 574 970 695 180
%e A375835 6 0 720 5858 16124 20240 11955 2700
%e A375835 7 0 5040 67752 285264 556591 559895 282555 56700
%e A375835 ...
%e A375835 The T(3, 2) = 8 chains in the poset of the permutations of {1, 2, 3} are:
%e A375835 {(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123),(1)(2)(3) < (132), (1)(23) < (123), (2)(13) < (132), (3)(12) < (123)}.
%p A375835 b := proc(n, k, t) option remember; if k < 0 then return 0 fi; if {n, k} = {0} then return 1 fi; add(ifelse(k = 1, 1, b(v, k - 1, 1))*abs(Stirling1(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); # _Peter Luschny_, Sep 05 2024
%t A375835 b[n_, k_, t_] := b[n, k, t] = If[k < 0, 0, If[n == 0 && k == 0, 1,
%t A375835 Sum[If[k == 1, 1, b[v, k - 1, 1]] * Abs[StirlingS1[n, v]], {v, k, n - t}]]];
%t A375835 T[n_, k_] := b[n, k, 0]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]
%Y A375835 Cf. A000007 (column k=0), A000142 (column k=1), A006472 (main diagonal), A375836 (row sums).
%Y A375835 Cf. A048994, A331955, A330804, A331956, A331957.
%K A375835 nonn,tabl
%O A375835 0,5
%A A375835 _Rajesh Kumar Mohapatra_, Aug 31 2024