On a Determinant of Teufl and Wagner Related to Set Partitions: Divisibility by the Catalan Power of det A

Teufl and Wagner posed the following problem (Problem 1 of their contribution to Oberwolfach Report 23/2018, pp. 1457–1458). Let \(A=(a_{ij})\) be an \(n\times n\) matrix of indeterminates, and let \(T=T(A)\) be the matrix, indexed by the set partitions of an \(n\)-set, whose entry \((P,Q)\) is the sum of the weights of the transitions from \(P\) to \(Q\); a transition is a set of edges between two copies \(X\), \(Y\) of the \(n\)-set which completes spanning trees of the blocks of \(P\) in \(X\) to a forest all of whose components meet \(Y\) and induce \(Q\) on \(Y\). Is \(\det T\) always divisible by \((\det A)^{C_n}\), where \(C_n\) is the \(n\)-th Catalan number? The report states \(\det T=(\det A)^{B_n}\) for \(n\le3\) (\(B_n\) the Bell number) and \(\det T=(\det A)^{14}\operatorname{per} A\) for \(n=4\), an experimental factorisation for \(n=5\) and experimental evidence for \(n=6\). We prove that the answer is yes for every \(n\): in \(\mathbb{Z}[a_{ij}]\) one has \(\det T=(\det A)^{C_n}R_n\), where \(R_n\) is the determinant of \(T(A)\) on an invariant subspace \(K\) of dimension \(B_n-C_n\) which does not depend on \(A\), and the map induced by \(T(A)\) on the quotient, which has the non-crossing partitions as a basis, has determinant exactly \((\det A)^{C_n}\). The proof uses the Grassmann-algebra representation of forests of Caracciolo, Sokal and Sportiello, in which \(T(A)\) becomes an integral operator \(G(A)\) with \(G(-A^{\mathsf{T}})\,G(A)=(-1)^n(\det A)^2\). These authors announced, without proof, that the image of the representation has the non-crossing partitions as a basis; we did not find their proof in print (the source file of their preprint contains a draft) and give one for the case needed. The subspace \(K\), the dimension and the planar basis of the quotient and the integrality of the projection are already contained in the work of Kenyon and Wilson on boundary partitions in trees, together with the meander determinant of Di Francesco, Golinelli and Guitter. The note adds the determinant on the quotient, hence the divisibility, the factorisation over \(\mathbb{Z}[a_{ij}]\), a proof of the formula for \(n=4\), and the exactness of the exponent \(C_n\) for \(4\le n\le8\) (computer-assisted for \(n\ge5\)). A second proof, found by one of the independent verification runs, uses the tree pairing of Kenyon and Wilson and the matrix-tree theorem. Factors of \(R_5\) and \(R_6\) are only tested; exactness for \(n\ge9\) and a description of \(R_n\) remain open. The wording of the UnsolvedMath record OWR-16164-020 mixes in the title of the next problem of the report. A literature search found no answer, which is not a proof of novelty; the divisibility is close to the published structure theory and may be known to specialists. This is an unrefereed note.