Geodesic
Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 217-229

Voir la notice de l'article provenant de la source Library of Science

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, Gl(n), of distance-l independent sets on the cycle Cn, studied by Comtet for l ≥ 0 and n ≥ 1 [sic]. We prove that values of Gl(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.
Keywords: distance independent set, Lucas numbers, Pisot numbers, power sums, generating functions, (co-) reciprocal polynomials 
@article{DMGT_2013_33_1_a17,
     author = {Skupie\'n, Zdzis{\l}aw},
     title = {Sums of {Powered} {Characteristic} {Roots} {Count} {Distance-Independent} {Circular} {Sets}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {217--229},
     publisher = {mathdoc},
     volume = {33},
     number = {1},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a17/}
}
TY  - JOUR
AU  - Skupień, Zdzisław
TI  - Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2013
SP  - 217
EP  - 229
VL  - 33
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a17/
LA  - en
ID  - DMGT_2013_33_1_a17
ER  - 
%0 Journal Article
%A Skupień, Zdzisław
%T Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets
%J Discussiones Mathematicae. Graph Theory
%D 2013
%P 217-229
%V 33
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a17/
%G en
%F DMGT_2013_33_1_a17
Skupień, Zdzisław. Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 217-229. http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a17/