Diagonalizability and symmetrizability of Sobolev-type bilinear forms: A combinatorial approach

In an earlier paper, Kwon, Littlejohn and Yoon characterized symmetric Sobolev bilinear forms and showed that they have, like symmetric matrices, a diagonal representation. In this paper, we present a new proof of one of their main results by interpreting the coefficients in the diagonal representation of a Sobolev-type bilinear form from a combinatorial point of view. We view this as an improvement over the original proof which relied on mathematical induction.