         - 초록검색Searching for abstracts            제출번호(No.) 0079 Poster Session Mathematics for Information Sciences (IS) Lattice path counting with a bounded strip restriction Kwangsoo Park1, Gi-Sang Cheon1Sungkyunkwan University1 We consider a path from the origin to the point $(dn,0)$ using the step set $S=\{(a,0),(b,1),(c,-1)\}$ for which the path cannot go below the line $y=-\alpha$ where $a,b,c\in \mathbb{N}$, $d:=gcd(a,b,c)$, $\alpha\in \mathbb{N}_0$. The number counted is denoted by $\phi_n^{(\alpha)}(a,b,c)$. This generalizes a classical counting problem. For instance, $\phi_n^{(0)}(1,1,1)$ and $\phi_{2n}^{(0)}(2,1,1)$ are the same as the $n$th Motzkin number and the $n$th Schr\"oder number, respectively. In this paper, we obtain the generating function for the numbers $\phi_n^{(\alpha)}(a,b,c)$. In particular, if $\alpha=0$ and $a=1$ then for any positive integers $b$ and $c$ we obtain $$\phi_n^{(0)}(1,b,c)=\sum_{k=0}^{\lfloor {n\over{b+c}} \rfloor} {{n-(b+c-2)k}\choose{2k}}c_k.$$ where $c_n$ denotes the $n$th Catalan number. Further we give a combinatorial proof for the following open questions posed by R. J. Mathar in 2012; \begin{eqnarray*} Q1.(n+3)a_n+(-2n-3)a_{n-1}+na_{n-2}+2(-2n+3)a_{n-3}=0 \end{eqnarray*} where $a_n=\phi_n^{(0)}(1,2,1)$. \begin{eqnarray*} Q2.(n+4)b_n+(n+4)b_{n-1}-(5n+8)b_{n-2}+3nb_{n-3}+4(2-n)b_{n-4}+12(3-n)a_{n-5}=0 \end{eqnarray*} where $b_n=\phi_n^{(0)}(1,3,1)$. The concept of Riordan matrix is extensively used throughout this paper. 05A15 lattice path, Riordan matrix Korean 게시판운영방침 | 개인정보취급방침 | 이메일주소무단수집거부 Copyright © Korean Mathematical Society. All Rights Reserved.  / TEL : 02-565-0361  / Fax : 02-565-0364 /  E-mail : kms@kms.or.kr 06130 서울 강남구 테헤란로7길 22 한국과학기술회관 신관 411호  / 사업자등록번호 : 105-82-04272 / 대표자 : 금종해