Abstract
Abstract. The r-ary number sequences given by(b (r)n ) n≥0 =1(r−1)n+ 1rnnare analogs of the sequence of the Catalan numbers 1n+1 2nn . Their his-tory goes back at least to Lambert [8] in 1758 and they are of considerableinterest in sequential testing. Usually, the sequences are considered sep-arately and the generalizations can go in several directions. Here we linkthe various r first by introducing a new combinatorial structure relatedto GR trees and then algebraically as well. This GR transition gener-alizes to give r-ary analogs of many sequences of combinatorial interest.It also lets us find infinite numbers of combinatorially defined sequencesthat lie between the Catalan numbers and the Ternary numbers, or moregenerally, between b (r)n and b (r+1)n . 1. IntroductionThe Catalan numbers C n = 1n+12nn are well known to occur in enumera-tion of a great variety of combinatorial objects. Some good resources for theCatalan numbers are provided by Stanley [15, 14], which include more than200 combinatorial interpretations.One of the important generalizations of the Catalan numbers are the r-arynumbers b
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