Some Identities for Palindromic Compositions Without $2$'s

Abstract:

      In this paper, we study the palindromic compositions of even integers when no $2$'s are allowed in a composition and its conjugate. We show that the number of these palindromes is equal to $2F_{n-1}$, where, $F_n$ is the $n$-th Fibonacci number. Consequently, we obtain several identities between the number of these palindromes, the number of compositions into parts equal to $1$'s or $2$'s, the number of compositions into odd parts and the number of compositions into parts greater than $1$.