Splitting subspaces of linear operators over finite fields

Splitting subspaces and a finite field interpretation of the Touchard-Riordan Formula

“…In the running example, the opening nodes are 1, 2, 3, 7 and the closing nodes are 4, 5, 6, 8. A crossing of the chord diagram is a pair of arcs (i, j), (k, l) such that i < k < j < l. The chord diagram above has two crossings, namely (1,4), (2,6) and (1,4), (3,5). Given a fixed-point-free involution σ, let v(σ) denote the number of crossings of its chord diagram.…”

“…The number of T -splitting subspaces is known when T has an irreducible characteristic polynomial [3,5,8], is regular nilpotent [2], is regular split semisimple [18,19], or when the invariant factors satisfy certain degree constraints [1]. In this article, we consider the case where d = 2.…”

Polynomial Matrices, Splitting Subspaces and Krylov Subspaces over Finite Fields

“…More generally, it is known [1,Cor. 3.7] that the number of splitting subspaces σ(m, d; T ) depends only on the similarity class type of T .…”

“…The case where T has an irreducible characteristic polynomial was settled by Chen and Tseng [6,Cor. 3.4] who proved a conjecture made in [10]; the case of cyclic nilpotent T has recently been settled in [1,Cor. 4.7].…”