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%I A357410 #18 Sep 26 2022 20:03:40
%S A357410 0,1,12,224,6960,397792,42001344,8547291008,3336917303040,
%T A357410 2565880599084544,3852698988517260288,11517943538435677485056,
%U A357410 67829192662051610706309120,799669932659456441970547744768,18652191511341505602408972738871296,873360272626100960024734923878091948032
%N A357410 a(n) is the number of covering relations in the poset P of n X n idempotent matrices over GF(2) ordered by A <= B if and only if AB = BA = A.
%C A357410 The order given for P in the title is equivalent to the ordering: A <= B if and only if image(A) is contained in image(B) and null(A) contains null(B).  Then A is covered by B if and only if there is a 1-dimensional subspace U such that image(B) = image(A) direct sum U.  If such a subspace U exists then it is unique and is equal to the intersection of null(A) with image(B).  The number of maximal chains in P is A002884(n).  The set of all idempotent matrices over GF(q) with this ordering is a binomial poset with factorial function |GL_n(F_q)|/(q-1)^n. (see Stanley reference).
%D A357410 R. P. Stanley, Enumerative Combinatorics, Vol I, second edition, page 320.
%H A357410 Wikipedia, <a href="http://en.wikipedia.org/wiki/Covering_relation">Covering relation</a>
%F A357410 Sum_{n>=0} a(n) x^n/B(n) = x * (Sum_{n>=0} x^n/B(n))^2 where B(n) = A002884(n).
%e A357410 a(2) = 12 because there are A132186(2) = 8 idempotent 2 X 2 matrices over GF(2).  The identity matrix covers 6 rank 1 matrices each of which covers the zero matrix for a total of 12 covering relations.  Cf. A296548.
%t A357410 nn = 15; B[q_, n_] := Product[q^n - q^i, {i, 0, n - 1}]/(q - 1)^n;
%t A357410 e[q_, u_] := Sum[u^n/B[q, n], {n, 0, nn}];Table[B[2, n], {n, 0, nn}] CoefficientList[Series[e[2, u] u e[2, u], {u, 0, nn}], u]
%Y A357410 Cf. A132186, A342245, A002884, A296548.
%K A357410 nonn
%O A357410 0,3
%A A357410 _Geoffrey Critzer_, Sep 26 2022

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