OFFSET

0,3

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).

REFERENCES

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

LINKS

Wikipedia, Covering relation

FORMULA

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

EXAMPLE

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]

CROSSREFS

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Sep 26 2022

STATUS

approved