On a combinatorial identity of Djakov and Mityagin (Q2831588)

As a by-product of the analysis of the spectrum of the Hill operator associated to a certain two-term trigonometric potential function, \textit{P. Djakov} and \textit{B. Mityagin} [J. Funct. Anal. 242, 157--194 (2007; Zbl 1115.34079)] proved that the following identity holds for all positive integers \(n\) and integers \(k\): NEWLINE\[NEWLINE\sum_{J \in {[n] \choose k}^\ast} \prod_{j \in J} j^2 = \sum_{J \in {[n] \choose k}^{\ast\ast}} \prod_{j \in J} j(n+1-j) \tag{1}NEWLINE\]NEWLINENEWLINE where \([n] := \left\{1,2,3,\ldots,n \right\}\) as usual, \({[n] \choose k}^\ast\) denotes the collection of \(k\)-subsets of \([n]\) all of whose elements are congruent to \(n\) modulo 2, and \({[n] \choose k}^{\ast\ast}\) denotes the collection of \(k\)-subsets of \([n]\) which do not contain two consecutive integers.NEWLINENEWLINEIdentity (1) appears in the enumeration of a selection of items in a pyramid composed of unit cubes. The first combinatorial proof of (1) was provided by \textit{D. Zagier} in the appendix to \textit{J. M. Borwein} et al. [Can. J. Math. 64, No. 5, 961--990 (2012; Zbl 1296.33011)].NEWLINENEWLINEIn the present note, the author establishes (1) in a completely new and natural way by calculating the characteristic polynomial and the eigenvalues of the so-called Kac matrix.