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A001833 Number of labeled graded partially ordered sets with n elements.

%I A001833 M3067 N1243 #41 Apr 01 2023 15:49:02
%S A001833 1,1,3,19,219,3991,106623,3964339,199515459,13399883551,1197639892983,
%T A001833 143076298623259,23053861370437659,5062745845287855271,
%U A001833 1530139311543346178223,641441466132460086890179,375107113287994040621904819,307244526491924695346004951151,353511145615118063468292270299943
%N A001833 Number of labeled graded partially ordered sets with n elements.
%C A001833 Here "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. Note that this notion of grading is weaker than in sequence A006860, which counts posets in which all maximal chains have the same length.
%D A001833 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001833 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001833 Andrew Howroyd, <a href="/A001833/b001833.txt">Table of n, a(n) for n = 0..100</a>
%H A001833 David A. Klarner, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80100-6">The number of graded partially ordered sets</a>, Journal of Combinatorial Theory, vol.6, no.1, pp.12-19, (January-1969).
%H A001833 D. A. Klarner, <a href="/A000798/a000798_11.pdf">The number of graded partially ordered sets</a>, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
%H A001833 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%e A001833 The poset on {a, b, c, d, e} defined by the relations a < b < c and d < e is counted by this sequence. (For example, one associated rank function is rk(a) = rk(d) = 0, rk(b) = rk(e) = 1 and rk(c) = 2.) However, the poset defined by the relations a < b < c and a < d < e < c is not graded and so not counted by this sequence.
%o A001833 (PARI) \\ C(n) is defined in A361951.
%o A001833 seq(n)={my(c=C(n)); Vec(serlaplace(c[n+1]/c[n]))} \\ _Andrew Howroyd_, Mar 31 2023
%Y A001833 Row sums of A361951.
%Y A001833 Graded posets with no chain of length 3 are counted by A001831.
%Y A001833 Cf. A223911, A228551, A361920 (unlabeled version).
%K A001833 nonn
%O A001833 0,3
%A A001833 _N. J. A. Sloane_
%E A001833 Corrected and edited by _Joel B. Lewis_, Mar 28 2011
%E A001833 a(7)-a(15) from _Daniele P. Morelli_, Aug 25 2013
%E A001833 a(16)-a(18) from _Sean A. Irvine_, Sep 25 2015