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A361951 Triangle read by rows: T(n,k) is the number of labeled weakly graded (ranked) posets with n elements and rank k.

%I A361951 #14 Apr 01 2023 14:26:14
%S A361951 1,0,1,0,1,2,0,1,12,6,0,1,86,108,24,0,1,840,2190,840,120,0,1,11642,
%T A361951 55620,31800,6840,720,0,1,227892,1858206,1428000,384720,60480,5040,0,
%U A361951 1,6285806,82938828,80529624,24509520,4626720,584640,40320
%N A361951 Triangle read by rows: T(n,k) is the number of labeled weakly graded (ranked) posets with n elements and rank k.
%C A361951 Here weakly graded means that there exists a rank function rk from the poset to the integers such that whenever v covers w in the poset, we have rk(v) = rk(w) + 1.
%C A361951 T(n,k) corresponds to a(k,n) = b(k,n) - b(k-1,n) in the Klarner reference. Figure 2 shows the posets of row n=4.
%H A361951 Andrew Howroyd, <a href="/A361951/b361951.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50).
%H A361951 D. A. Klarner, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80100-6">The number of graded partially ordered sets</a>, J. Combin. Theory, 6 (1969), 12-19.
%H A361951 Wikipedia, <a href="https://en.wikipedia.org/wiki/Graded_poset">Graded poset</a>.
%F A361951 E.g.f. of column k >=2: C(k,x)/C(k-1,x) - C(k-1,x)/C(k-2,x) where C(k,x) is the e.g.f. of column k of A361950.
%e A361951 Triangle begins:
%e A361951   1;
%e A361951   0, 1;
%e A361951   0, 1,      2;
%e A361951   0, 1,     12,       6;
%e A361951   0, 1,     86,     108,      24;
%e A361951   0, 1,    840,    2190,     840,    120;
%e A361951   0, 1,  11642,   55620,   31800,   6840,   720;
%e A361951   0, 1, 227892, 1858206, 1428000, 384720, 60480, 5040;
%e A361951   ...
%o A361951 (PARI) \\ Here C(n) gives columns of A361950 as vector of e.g.f.'s.
%o A361951 S(M)={matrix(#M, #M, i, j, sum(k=0,  i-j, 2^((j-1)*k)*M[i-j+1,k+1])/(j-1)! )}
%o A361951 C(n,m=n)={my(M=matrix(n+1, n+1), c=vector(m+1), A=O(x*x^n)); M[1, 1]=1; c[1]=1+A; for(h=1, m, M=S(M); c[h+1]=sum(i=0, n, vecsum(M[i+1, ])*x^i, A)); c}
%o A361951 T(n)={my(c=C(n), b=vector(n+1, h, c[h]/c[max(h-1,1)])); Mat(vector(n+1, h, Col(serlaplace(b[h]-if(h>1, b[h-1])), -n-1)))}
%o A361951 { my(A=T(7)); for(n=1, #A, print(A[n, 1..n])) }
%Y A361951 Row sums are A001833.
%Y A361951 Column k=2 is A055531.
%Y A361951 Partial row sums include A000007, A000012, A001831, A001832.
%Y A361951 Main diagonal is A000142.
%Y A361951 The unlabeled version is A361953.
%Y A361951 Cf. A342587, A361950.
%K A361951 nonn,tabl
%O A361951 0,6
%A A361951 _Andrew Howroyd_, Mar 31 2023