On the combinatorics of polynomial generalizations of Rogers-Ramanujan-type identities (Q1613561)

The author considers the polynomial sequence \(\{P_n(q)\}\), defined inductively by \(P_0(q)=1\), \(P_1(q)= 1+q\), \(P_n(q)= (1-q^2+q^{2n-1}) P_{n-1}(q)+ q^2P_{n-2}(q)\) for \(n\geq 2\). (Setting \(q=1\) yields the Fibonacci sequence.) In addition, the author defines ``Frobenius even alternating'' partitions, and shows that the number of such partitions that are self-conjugate and have largest part at most \(n\) is \(F_n\) (the \(n\)th Fibonacci number).