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

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.

Cf. A342587, A361950.

Sequence in context: A091803 A123002 A261161 * A137514 A367381 A322221

Adjacent sequences: A361948 A361949 A361950 * A361952 A361953 A361954

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 31 2023

Status

approved