A359313 Triangular array read by rows. T(n,k) is the number of Green's H-classes contained in the D-class of rank k matrices in the semigroup Mat_n(F_2) of n X n matrices over the field F_2. n>=0, 0<=k<=n.

1, 1, 1, 1, 9, 1, 1, 49, 49, 1, 1, 225, 1225, 225, 1, 1, 961, 24025, 24025, 961, 1, 1, 3969, 423801, 1946025, 423801, 3969, 1, 1, 16129, 7112889, 139499721, 139499721, 7112889, 16129, 1, 1, 65025, 116532025, 9439094025, 40315419369, 9439094025, 116532025, 65025, 1

Offset: 0

Author

Geoffrey Critzer, Dec 25 2022

Keywords

• nonn
• tabl

Comments

For all a,b in the semigroup Mat_n(F_2), aDb if and only if rank(a)=rank(b). Also, aHb if and only if the row(a)=row(b) and col(a)=col(b). So the H classes correspond to ordered pairs (U,W) of subspaces of F_2^n with dim(U) = dim(W). Let a in Mat_n(F_2) be such that col(a) = U and row(a)=W. The size of H_a, the H-class containing a is |GL_d(F_2)| where d=dim(U). H_a contains an idempotent if and only if col(a) + perp(row(a)) is a direct sum decomposition of F_2^n where perp(X)={v in F_2^n: v*x = 0 for all x in X}.

Let H_a,H_b be H-classes in Mat_n(F_2). Let H_a ~ H_b if and only if col(a) is contained in col(b) and row(a) is contained in row(b). Then ~ is a partial order relation on the set of all H-classes in Mat_n(F_q). The poset is isomorphic to a typical n-interval in the binomial poset L*L where L is the binomial poset of all finite dimensional subspaces over a countably infinite dimensional subspace and * is the Segre product (see Stanley reference). T(n,k) is the number of elements at rank k in an n-interval of L*L.

Examples

			 1
 1,   1
 1,   9,     1
 1,  49,    49,     1
 1, 225,  1225,   225,   1
 1, 961, 24025, 24025, 961, 1
		

References

• R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, section 3.18.

Links

• Wikipedia, Green's relations

Crossrefs

Cf. A243950 (row sums), A022166, A005329, A002884, A002416, A296548 (a subposet of L*L).

Programs

Mathematica

nn = 8; B[n_, q_] := QFactorial[n, q]^2; e[x_] := Sum[x^n/B[n, 2], {n, 0, nn}]; Map[Select[#, # > 0 &] &, Table[QFactorial[n, 2]^2, {n, 0, nn}] CoefficientList[
   Series[e[x] e[y x], {x, 0, nn}], {x, y}]]

Formula

T(n,k) = A022166(n,k)^2.

Sum_{k=0..n} T(n,k)*A002884(k) = A002416(n).

Let B(n) = A005329(n)^2. Let E(x)=Sum_{n>=0}x^n/B(n). Then Sum{n>=0} Sum{k=0..n} T(n,k)*y^k*x^n/B(n) = E(x)*E(y*x).