Cyclic, Dihedral and Symmetrical Carlitz Compositions of a Positive Integer
9 Ikarou Str.
A linear composition of a positive integer N is a list of positive
integers (called parts) whose sum equals N. We distinguish two kinds of
cyclic compositions, which we call C-type and CR-type. A CR-type cyclic
composition of N is an equivalence class of all linear compositions of
N that can be obtained from each other by a cyclic shift, while a
dihedral composition is an equivalence class of all linear compositions
of N that can be obtained from each other by a cyclic shift or a
reversal of order. A linear Carlitz composition is one where adjacent
parts are distinct. A C-type cyclic Carlitz composition is a linear
Carlitz composition whose first and last parts are distinct, whereas a
CR-type cyclic Carlitz composition is an equivalence class of C-type
Carlitz compositions that can be obtained from each other by a cyclic
shift. We distinguish two kinds of linear palindromic compositions
(type I and type II). We derive generating functions for the number of
type II linear palindromic Carlitz compositions, and we provide a new
proof of a result by J. Taylor about C-type Carlitz compositions. Using
these results, we derive formulas about CR-type Carlitz compositions,
symmetrical CR-type compositions, and dihedral Carlitz compositions.
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(Concerned with sequences
Received February 1 2017; revised versions received February 12 2017; July 12 2017; August 24 2017.
Published in Journal of Integer Sequences, September 2 2017.
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