A006860 Erroneous version of A223911: Tiered orders on n nodes.

1, 3, 13, 111, 1381, 25623, 678133, 26269735, 1447451707, 114973020921, 13034306495563

Offset: 1

Author

Simon Plouffe

Keywords

• dead

Comments

WARNING: The currently listed value of a(8) is inconsistent with the result from Kreweras and Klarner quoted below, as pointed out by Michel Marcus. - M. F. Hasler, Nov 03 2012

A corrected version of this sequence is A223911. - Joerg Arndt, Mar 29 2013

Graded posets, i.e., those in which every maximal chain has the same length. (The terminology "graded" is also used to refer to a weaker notion; see A001833.)

Kreweras observed and Klarner proved that a(n) is congruent to 1 (resp. 3) modulo 6 when n is odd (resp. even). - Michel Marcus, Nov 03 2012

Using the formulas in the paper from Klarner (cf. PARI code), I get 1, 3, 13, 85, 801, 10231, 168253, 3437673, 85162465, 2511412651, 86805640461, 3469622549053, ... - M. F. Hasler, Nov 07 2012

The values currently in the sequence through 25623 are certainly correct (I've enumerated these posets by brute force and other methods). (...) Klarner's eq.(2) contains a typo: instead of f(m_1, m_h) it should be f(m_1, m_2). (The point here is that the Hasse diagram of each of these posets decomposes as a bunch of bipartite graphs layered on top of each other; there are f(m_1, m_2) ways to choose the bipartite graph between the first two ranks of vertices, then f(m_2, m_3) ways to choose the bipartite graph between the second and third ranks of vertices, etc.) (...). When I implement Klarner's eqs.(1) and (2) (corrected) I get the following sequence: 1, 3, 13, 111, 1381, 25623, 678133, 26169951, 1447456261, 114973232583, ... Now we get the right terms up as far as I personally have experience (...) and they agree with Kreweras (and the current OEIS sequence) until a(8), at which point there is disagreement. - Joel B. Lewis, Mar 06 2013; private communication to M. F. Hasler

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• D. Klarner, The number of tiered posets modulo six, Discrete Math., 62 (1986), 295-297.
• G. Kreweras, Dénombrement des ordres étagés, Discrete Math., 53 (1985), 147-149.

Programs

PARI

ee(n)={my(f(m,n)=sum(k=0,m,(-1)^(m-k)*binomial(m,k)*(2^k-1)^n), C(n,m)=n!/prod(i=1,#m,m[i]!), t(h,n)=my(s=0); forvec(m=vector(h,i,[if(iM. F. Hasler, Nov 07 2012

Extensions

Error in a(8) pointed out by Michel Marcus, Nov 03 2012