On The Maximum Jump Number $M(2k-1,k)$

Abstract

If $n$ and $k$ ($n\geq k$) are large enough , it is   quite difficult to give the value of $M(n,k)$. R.A. Brualdi and H.C. Jung gave   a table about the value of $M(n,k)$ for $1\leq k \leq n\leq 10 $. In this paper, we show that   $4(k-1)-\lceil\sqrt{k-1}\rceil\leq M(2k-1,k)\leq 4k-7$ holds for   $k\geq 6$. Hence, $M(2k-1,k)=4k-7$ holds for $6\leq k \leq 10$,   which verifies that their conjecture   $M(2k+1,k+1)=4k-\lceil\sqrt{k}\rceil$ holds for $5\leq k\leq 9$,   and disprove their conjecture   $M(n,k)<M(n+l_{1},k+l_{2})$ for $l_{1}= 1 $, $l_{2}= 1 $.