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The $r$-ary number sequences given by $$(\mathfrak{b}_{n}^{(r)})_{n\ge0} ={\frac{1}{(r-1)n+1}}{\binom{rn}{n}}$$ are analogs of the sequence of the Catalan numbers ${\frac{1}{n+1}}{\binom{2n}{n}}$. Their history goes back at least to Lambert \cite{BLam} in 1758 and they are of considerable interest in sequential testing. Usually, the sequences are considered separately and the generalizations can go in several directions. Here we link the various $r$ first by introducing a new combinatorial structure related to GR trees and then algebraically as well. This GR transition generalizes to give $r$-ary analogs of many sequences of combinatorial interest. It also lets us find infinite numbers of combinatorially defined sequences that lie between the Catalan numbers and the Ternary numbers, or more generally, between $\mathfrak{b}_n^{(r)}$ and $\mathfrak{b}_n^{(r+1)}$.
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참고문헌 (15)
참고문헌 신청
D. Davenport / 2012 / The oldest child tree / Congr. Numer 213:123~131
N. Dershowitz / 1980 / Enumerations of ordered trees / Discrete Math 31(1):9~28
E. Deutsch / 2004 / Ordered trees with prescribed root degrees, node degrees, and branch lengths / Discrete Math 282(1-3):89~94
R. Donaghey / 1977 / Motzkin numbers / J. Combin. Theory Ser. A 23(3):291~301
P. Flajolet / 2009 / Analytic Combinatorics / Cambridge University Press
N. Dershowitz / 1980 / Enumerations of ordered trees / Discrete Math 31(1):9~28
E. Deutsch / 2004 / Ordered trees with prescribed root degrees, node degrees, and branch lengths / Discrete Math 282(1-3):89~94
R. Donaghey / 1977 / Motzkin numbers / J. Combin. Theory Ser. A 23(3):291~301
P. Flajolet / 2009 / Analytic Combinatorics / Cambridge University Press
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