Some combinatorial identities

If j = 2 in (C6) , we obtain (3.7) a 2 001 + a 2 011 + a QQ1 a Q11 = 1. Using (3.6) and (3.7) along with the fact that a 0 0 1 = 1, we see that # 0 1 1 = 0. Since all the variables in (C7) have already been uniquely determined, we proceed to (C8), where we obtain c 2-~ 2 and (3.9) a 2 , + 2 + a a = i, (3.8) a 2 0Q1 + a 1 0 1 + 2 a 0 0 1 a 1 0 1 = 1 so that a 101 = 0. From (C10), we obtain and (3.11) a 2 010 + al lQ + a 0 1 0 a 1 1 0 = 1, so that a 110-0. From (C12) , we obtain, after simplification, (3.12) al ±1 + 2a l l x = 0 and (3.13) a 2 11 + a 1±1 = 0, so that a 111 = 0. We have now uniquely determined all 27 coefficients in (3.1). Thus, is the only reduced local permutation polynomial in three variables over Z 3 and, hence, there is precisely one reduced Latin cube of order three. If we list the cube in terms of the three Latin squares of order three which form its different levels, we can list the only reduced Latin cube of order three as 012 120 201 120 201 012 201 012 120. In this paper, we wish to derive some combinatorial identities (partly known, partly apparently new) by combining well-known recurrence relations with known forms for characteristic polynomials of paths and cycles (i.e., of their adjacency matrices). We also obtain some extensions of known results .