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• W1633532382 abstract "Abstract: Cayley's 1875 enumerations of and alkanes (unlabeled trees ofvalency at most 4) are corrected and extended --- possibly for the first time in 124 years. 1. Introduction In 1875 Cayley attempted to enumerate alkanes C n H 2n+2 , or equivalently n-node unlabeled trees inwhich each node has degree at most 4, and published a short note [Cay75] containing the table: n 1 2 3 4 5 6 7 8 9 10 11 12 13 1 0 1 1 2 2 6 9 20 37 86 183419 (1) 0 1 0 1 1 3 3 9 15 38 73 174380 (2) total 1 1 1 2 3 5 9 18 35 75159357799 (3)(The terms centered and bicentered are defined below.) This table was reproduced by Busackerand Saaty in 1965 [BuS65], and the three sequences were included in [HIS].In fact the last two columns are in error, as had already been pointed out by Herrmann in 1880[Her80]. Herrmann uses a different method from Cayley, and gives the correct values 355 (for n=12)and 802 (for n=13) for sequence (3). However, neither in [Her80] nor in his two later notes [Her97],[Her98] does he mention sequences (1) and (2).The alkane sequence (3) is also discussed in the works by Schiff [Sch75], Losanitsch [Los97],[Los97a], Henze and Blaire [HeB31], Perry [Per32], Polya [Polya36], [Polya37], Harary and Norman[HaN60], Lederberg [Led69], Read [Rea76], Robinson, Harary and Balaban [RoHB76], andBergeron, Labelle and Leroux [BeLL98]. The simplest generating function is due to Harary andNorman (see Section 4 of [Rea76] or p. 289 of [BeLL98]). However, none of these authors useCayley's method, and as far as we can tell none of them discuss sequences (1) and (2).In 1988 R. K. Guy wrote to N.J.A.S., pointing out that there were errors in these three sequences, andsuggested that Polya counting theory be used to extend (1) and (2). (The correct version of (3),sequence A000602, was already present in [HIS].) To do so is the goal of the present note.We confess to having another, more ignoble reason for wishing to extend sequences (1) and (2). Thesequences in the data-base [EIS] are numbered (01, 02, 03, ...), and several people" @default.
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• W1633532382 date "1999-01-01" @default.
• W1633532382 modified "2023-10-16" @default.
• W1633532382 title "On Cayley's Enumeration of Alkanes (or 4-Valent Trees)" @default.
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