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A222863 Strongly graded (3+1)-free partially ordered sets (posets) on n labeled vertices. 3
1, 1, 3, 13, 111, 1381, 22383, 461413, 12163791, 420626821, 19880808303, 1337330559973, 130909732781391, 18649561895661061, 3830195104867879023, 1124247654215697637093, 469367653568553278229711, 278046313987470874905216901, 233462156432002170491075384943 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Here "strongly graded" means that every maximal chain has the same length. Alternate terminology includes "graded" (e.g., in Stanley 2012) and "tiered" (as in A006860). A poset is said to be (3+1)-free if it does not contain four elements a, b, c, d such that a < b < c and d is incomparable to the other three.

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Volume 1.  Cambridge University Press.  2nd edition, 2012.  http://math.mit.edu/~rstan/ec/ec1/

LINKS

Table of n, a(n) for n=0..18.

J. B. Lewis and Y. X. Zhang, Enumeration of Graded (3+1)-Avoiding Posets, J. Combin. Theory Ser. A 120 (2013), no. 6, 1305-1327.

FORMULA

G.f.: H(e^x, Psi(x)) where H(x, y) = 1 + (2x^3 - 3x^2 + (x^3 - 4x^2 + 4x)y)/(2x^2 + x + (x^2 - 2x - 1)y) and Psi(x) is the g.f. for A047863.

MATHEMATICA

m = maxExponent = 19;

Psi[x_] = Sum[E^(2^n*x)*x^n/n!, {n, 0, m}] + O[x]^m;

H[x_, y_] = 1+(2x^3 - 3x^2 + (x^3 - 4x^2 + 4x)y)/(2x^2 + x + (x^2-2x-1) y);

CoefficientList[H[E^x, Psi[x]] + O[x]^m, x] Range[0, m-1]! (* Jean-Fran├žois Alcover, Dec 11 2018 *)

CROSSREFS

For strongly graded (3+1)-free posets by height, see A222864. For weakly graded (3+1)-free posets, see A222865. For all strongly graded posets, see A006860. For all (3+1)-free posets, see A079145.

Sequence in context: A061377 A183604 A228563 * A223911 A006860 A181083

Adjacent sequences:  A222860 A222861 A222862 * A222864 A222865 A222866

KEYWORD

nonn

AUTHOR

Joel B. Lewis, Mar 07 2013

STATUS

approved

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Last modified February 22 17:35 EST 2019. Contains 320400 sequences. (Running on oeis4.)