OFFSET
0,4
COMMENTS
For a transitive relation R on [n], let E = domain(R intersect R^(-1)) and let F = [n]\E. Let q(R) = R intersect E X E and let s(R) = R intersect F X F. Let ~ be the equivalence relation on the set of transitive binary relations on [n] defined by: R_1 ~ R_2 iff q(R_1) = q(R_2) and s(R_1) = s(R_2). Here, two transitive relations are inequivalent if they are in distinct equivalence classes under ~. q(R) is a quasi-order (A000798) and s(R) is a strict partial order (A001035). The relation q(R) union s(R) may be taken as its class representative. See Norris link.
LINKS
E. Norris, The structure of an idempotent relation, Semigroup Forum, Vol 18 (1979), 319-329.
FORMULA
E.g.f.: p(exp(y*x) - 1)*p(x) where p(x) is the e.g.f. for A001035.
EXAMPLE
Triangle begins
1;
1, 1;
3, 2, 4;
19, 9, 12, 29;
219, 76, 72, 116, 355;
4231, 1095, 760, 870, 1775, 6942;
...
MATHEMATICA
nn = 8; posets = Select[Import["https://oeis.org/A001035/b001035.txt", "Table"],
Length@# == 2 &][[All, 2]]; p[x_] := Total[posets Table[x^i/i!, {i, 0, 18}]];
Map[Select[#, # > 0 &] &, Table[n!, {n, 0, nn}] CoefficientList[Series[ p[Exp[ y x] - 1]*p[ x], {x, 0, nn}], {x, y}]] // Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jan 31 2024
STATUS
approved