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A361956 Triangle read by rows: T(n,k) is the number of labeled tiered posets with n elements and height k.

%I A361956 #11 Apr 03 2023 21:09:34
%S A361956 1,0,1,0,1,2,0,1,6,6,0,1,50,36,24,0,1,510,510,240,120,0,1,7682,10620,
%T A361956 4800,1800,720,0,1,161406,312606,136920,47040,15120,5040,0,1,4747010,
%U A361956 13439076,5630184,1678320,493920,141120,40320,0,1,194342910,821218110,319384800,83963880,21137760,5594400,1451520,362880
%N A361956 Triangle read by rows: T(n,k) is the number of labeled tiered posets with n elements and height k.
%C A361956 A tiered poset is a partially ordered set in which every maximal chain has the same length.
%H A361956 Andrew Howroyd, <a href="/A361956/b361956.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50).
%e A361956 Triangle begins:
%e A361956   1;
%e A361956   0, 1;
%e A361956   0, 1,      2;
%e A361956   0, 1,      6,      6;
%e A361956   0, 1,     50,     36,     24;
%e A361956   0, 1,    510,    510,    240,   120;
%e A361956   0, 1,   7682,  10620,   4800,  1800,   720;
%e A361956   0, 1, 161406, 312606, 136920, 47040, 15120, 5040;
%e A361956   ...
%o A361956 (PARI)
%o A361956 S(M)={my(N=matrix(#M-1, #M-1, i, j, sum(k=1, i-j+1, (2^j-1)^k*M[i-j+1, k])/j!)); for(i=1, #N, for(j=1, i, N[i,j] -= sum(k=1, j-1, N[i-k, j-k]/k!))); N}
%o A361956 C(n)={my(M=matrix(n+1,n+1), R=M); M[1,1]=R[1,1]=1; for(h=1, n, M=S(M); for(i=h, n, R[i+1,h+1] = i!*vecsum(M[i-h+1,]))); R}
%o A361956 { my(A=C(7)); for(i=1, #A, print(A[i, 1..i])) }
%Y A361956 Row sums are A223911.
%Y A361956 Column k=2 is A052332.
%Y A361956 Main diagonal is A000142.
%Y A361956 The unlabeled version is A361957.
%Y A361956 Cf. A222864, A361951.
%K A361956 nonn,tabl
%O A361956 0,6
%A A361956 _Andrew Howroyd_, Apr 02 2023