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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.
5
1, 0, 1, 0, 1, 2, 0, 1, 12, 6, 0, 1, 86, 108, 24, 0, 1, 840, 2190, 840, 120, 0, 1, 11642, 55620, 31800, 6840, 720, 0, 1, 227892, 1858206, 1428000, 384720, 60480, 5040, 0, 1, 6285806, 82938828, 80529624, 24509520, 4626720, 584640, 40320
OFFSET
0,6
COMMENTS
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.
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50).
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
Wikipedia, Graded poset.
FORMULA
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.
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1, 2;
0, 1, 12, 6;
0, 1, 86, 108, 24;
0, 1, 840, 2190, 840, 120;
0, 1, 11642, 55620, 31800, 6840, 720;
0, 1, 227892, 1858206, 1428000, 384720, 60480, 5040;
...
PROG
(PARI) \\ Here C(n) gives columns of A361950 as vector of e.g.f.'s.
S(M)={matrix(#M, #M, i, j, sum(k=0, i-j, 2^((j-1)*k)*M[i-j+1, k+1])/(j-1)! )}
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}
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)))}
{ my(A=T(7)); for(n=1, #A, print(A[n, 1..n])) }
CROSSREFS
Row sums are A001833.
Column k=2 is A055531.
Partial row sums include A000007, A000012, A001831, A001832.
Main diagonal is A000142.
The unlabeled version is A361953.
Sequence in context: A123002 A261161 A387648 * A137514 A367381 A322221
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Mar 31 2023
STATUS
approved