No CrossRef data available.
Article contents
BASE SHIFTING SEQUENCES AND THE PALINDROME PROPERTY
Part of:
Combinatorics
Published online by Cambridge University Press: 10 January 2025
Abstract
For 
$M,N,m\in \mathbb {N}$ with 
$M\geq 2, N\geq 1$ and 
$m \geq 2$, we define two families of sequences, 
$\mathcal {B}_{M,N}$ and 
$\mathcal {B}_{M,N,m}$. The nth term of 
$\mathcal {B}_{M,N}$ is obtained by expressing n in base M and recombining those digits in base N. The nth term of 
$\mathcal {B}_{M,N,m}$ is defined by 
$\mathcal {B}_{M,N,m}(n):=\mathcal {B}_{M,N}(n) \; (\textrm {mod}\;m)$. The special case 
$\mathcal {B}_{M,1,m}$, where 
$N=1$, yields the digit sum sequence 
$\mathbf {t}_{M,m}$ in base M mod m. We prove that 
$\mathcal {B}_{M,N}$ is the fixed point of a morphism 
$\mu _{M,N}$ at letter 
$0$, similar to a property of 
$\mathbf {t}_{M,m}$. Additionally, we show that 
$\mathcal {B}_{M,N,m}$ contains arbitrarily long palindromes if and only if 
$m=2$, mirroring the behaviour of the digit sum sequence. When 
$m\geq M$ and N, m are coprime, we establish that 
$\mathcal {B}_{M,N,m}$ contains no overlaps.
MSC classification
Primary:
05A05: Permutations, words, matrices
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
References
Allouche, J.-P. and Shallit, J., ‘The ubiquitous Prouhet–Thue–Morse sequence’, in: Sequences and their Applications, Discrete Mathematics and Theoretical Computer Science (eds. Ding, C., Helleseth, T. and Niederreiter, H.) (Springer, London, 1998), 1–16.Google Scholar
Allouche, J.-P. and Shallit, J., ‘Sums of digits, overlaps and palindromes’, Discrete Math. Theor. Comput. Sci. 4 (2000), 1–10.Google Scholar
Allouche, J.-P. and Shallit, J., Automatic Sequences: Theory, Applications, Generalizations (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Berstel, J., Lauve, A., Reutenauer, C. and Saliola, F., Combinatorics on Words: Christoffel Words and Repetition in Words, CRM Monograph Series, 27 (American Mathematical Society, Providence, RI, 2008).CrossRefGoogle Scholar
Bolker, E., Offner, C., Richman, R. and Zara, C., ‘The Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences’, J. Comb. 7(1) (2016), 117–133.Google Scholar
Cusick, T. W. and Ciungu, L. C., ‘Sum of digits sequences modulo m
’, Theor. Comput. Sci. 412(35) (2011), 4738–4741.CrossRefGoogle Scholar
Thue, A., ‘Über unendliche Zeichenreihen,’ Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse 7 (1906), 22 pages.Google Scholar
Thue, A., Selected Mathematical Papers of Axel Thue (eds. Nagell, T., Selberg, A., Selberg, S. and Thalberg, K.) (Universitetsforlaget, Oslo, 1977).Google Scholar