Sums of powered characteristic roots count distance-independent circular sets
Zdzisław Skupień · Discussiones Mathematicae Graph Theory · 2012
Significant values of a combinatorial count need not fit the recurrence for the count.Consequently, initial values of the count can much outnumber those for the recurrence.So is the case of the count, G l (n), of distance-l independent sets on the cycle C n , studied by Comtet for l ≥ 0 and n ≥ 1 [sic].We prove that values of G l (n) are nth power sums of the characteristic roots of the corresponding recurrence unless 2 ≤ n ≤ l.Lucas numbers L(n) are thus generalized since L(n) is the count in question if l = 1.Asymptotics of the count for 1 ≤ l ≤ 4 involves the golden ratio (if l = 1) and three of the four smallest Pisot numbers inclusive of the smallest of them, plastic number, if l = 4.It is shown that the transition from a recurrence to an OGF, or back, is best presented in terms of mutually reciprocal (shortly: coreciprocal) polynomials.Also the power sums of roots (i.e., moments) of a polynomial have the OGF expressed in terms of the co-reciprocal polynomial.