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제출번호(No.) 0079
분류(Section) Poster Session
분과(Session) Mathematics for Information Sciences (IS)
영문제목
(Title(Eng.))
Lattice path counting with a bounded strip restriction
저자(Author(s))
Kwangsoo Park1, Gi-Sang Cheon1
Sungkyunkwan University1
초록본문(Abstract) 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.
분류기호
(MSC number(s))
05A15
키워드(Keyword(s)) lattice path, Riordan matrix
강연 형태
(Language of Session (Talk))
Korean

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