Let $T$ be a linear endomorphism of a $2m$-dimensional vector space. An $m$-dimensional subspace $W$ is said to be $T$-splitting if $W$ intersects $TW$ trivially.

When the underlying field is finite of order $q$ and $T$ is diagonal with distinct eigenvalues, the number of splitting subspaces is essentially the the generating function of chord diagrams weighted by their number of crossings with variable $q$. This generating function was studied by Touchard in the context of the stamp folding problem. Touchard obtained a compact form for this generating function, which was explained more clearly by Riordan.

We provide a formula for the number of splitting subspaces for a general operator $T$ in terms of the number of $T$-invariant subspaces of various dimensions. Specializing to diagonal matrices with distinct eigenvalues gives an unexpected and new proof of the Touchardâ€“Riordan formula.

This is based on joint work with Samrith Ram.

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Last updated: 06 Nov 2024