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.

0, 1, 12, 224, 6960, 397792, 42001344, 8547291008, 3336917303040, 2565880599084544, 3852698988517260288, 11517943538435677485056, 67829192662051610706309120, 799669932659456441970547744768, 18652191511341505602408972738871296, 873360272626100960024734923878091948032

Offset: 0

Author

Geoffrey Critzer, Sep 26 2022

Keywords

• nonn

Comments

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).

Examples

			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.
		

References

• R. P. Stanley, Enumerative Combinatorics, Vol I, second edition, page 320.

Links

• Wikipedia, Covering relation

Crossrefs

Cf. A132186, A342245, A002884, A296548.

Programs

Mathematica

nn = 15; B[q_, n_] := Product[q^n - q^i, {i, 0, n - 1}]/(q - 1)^n;
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]

Formula

Sum_{n>=0} a(n) x^n/B(n) = x * (Sum_{n>=0} x^n/B(n))^2 where B(n) = A002884(n).