### On the Numbers of Palindromes

Bang, Sejeong;Feng, Yan-Quan;Lee, Jaeun

• 투고 : 2015.07.27
• 심사 : 2015.12.23
• 발행 : 2016.06.23
• 4 1

#### 초록

For any integer $n{\geq}2$, each palindrome of n induces a circulant graph of order n. It is known that for each integer $n{\geq}2$, there is a one-to-one correspondence between the set of (resp. aperiodic) palindromes of n and the set of (resp. connected) circulant graphs of order n (cf. [2]). This bijection gives a one-to-one correspondence of the palindromes ${\sigma}$ with $gcd({\sigma})=1$ to the connected circulant graphs. It was also shown that the number of palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ is the same number of aperiodic palindromes of n. Let $a_n$ (resp. $b_n$) be the number of aperiodic palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ (resp. $gcd({\sigma}){\neq}1$). Let $c_n$ (resp. $d_n$) be the number of periodic palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ (resp. $gcd({\sigma}){\neq}1$). In this paper, we calculate the numbers $a_n$, $b_n$, $c_n$, $d_n$ in two ways. In Theorem 2.3, we $n_d$ recurrence relations for $a_n$, $b_n$, $c_n$, $d_n$ in terms of $a_d$ for $d{\mid}n$ and $d{\neq}n$. Afterwards, we nd formulae for $a_n$, $b_n$, $c_n$, $d_n$ explicitly in Theorem 2.5.

#### 키워드

compositions of n;palindromes;circulant graphs

#### 참고문헌

1. H. Baek, The number of aperiodic palindromes of n ${\leq}$ 127, available at http://chaos.cu.ac.kr/-tnad/.
2. H. Baek. S. Bang, D. Kim and J. Lee, A bijection between aperiodic palindromes and connected circulant graphs, preprint.
3. S. Heuhach, P. Chinn and R. Grimaldi, Rises, Levels, Drops and "+" Signs in Compositions: Extensions of a Paper by Alladi and Hoggatt, Fibonacci Quarterly 41(3)(1975), 229-239.
4. N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A179519.

#### 과제정보

연구 과제 주관 기관 : National Natural Science Foundation of China, National Research Foundation of Korea(NRF)