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A361953 Triangle read by rows: T(n,k) is the number of unlabeled weakly graded (ranked) posets with n elements and rank k.

%I A361953 #12 Apr 01 2023 14:26:28
%S A361953 1,0,1,0,1,1,0,1,3,1,0,1,8,6,1,0,1,20,30,9,1,0,1,55,145,66,12,1,0,1,
%T A361953 163,745,465,111,15,1,0,1,556,4245,3444,964,165,18,1,0,1,2222,27880,
%U A361953 28024,8618,1652,228,21,1,0,1,10765,218058,259974,83322,16569,2556,300,24,1
%N A361953 Triangle read by rows: T(n,k) is the number of unlabeled weakly graded (ranked) posets with n elements and rank k.
%C A361953 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.
%H A361953 Andrew Howroyd, <a href="/A361953/b361953.txt">Table of n, a(n) for n = 0..860</a> (rows 0..40)
%H A361953 Andrew Howroyd, <a href="/A361953/a361953.gp.txt">PARI Program</a>, Apr 2023.
%H A361953 Wikipedia, <a href="https://en.wikipedia.org/wiki/Graded_poset">Graded poset</a>.
%F A361953 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 g.f. of column k of A361952.
%e A361953 Triangle begins:
%e A361953   1;
%e A361953   0, 1;
%e A361953   0, 1,   1;
%e A361953   0, 1,   3,    1;
%e A361953   0, 1,   8,    6,    1;
%e A361953   0, 1,  20,   30,    9,   1;
%e A361953   0, 1,  55,  145,   66,  12,   1;
%e A361953   0, 1, 163,  745,  465, 111,  15,  1;
%e A361953   0, 1, 556, 4245, 3444, 964, 165, 18, 1;
%e A361953   ...
%o A361953 (PARI) \\ See link for program code.
%o A361953 { my(A=A361953tabl(8)); for(i=1, #A, print(A[i, 1..i])) }
%Y A361953 Row sums are A361920.
%Y A361953 The labeled version is A361951.
%Y A361953 Cf. A263859, A361952, A361954 (connected).
%K A361953 nonn,tabl
%O A361953 0,9
%A A361953 _Andrew Howroyd_, Mar 31 2023