How to prove or disprove this statement:
For all $c<z<0<s$, there exists $0<k\leq i$, $0\leq j<s+i$, such that all conditions hold simultaneously:
- $z=is+i(i-1)/2-j-k(s+i+2j)-k(k-1)/2$,
- $z<c+s+i+2j+k+1$ and $0<is+i(i-1)/2-j-(k-1)(s+i+2j)-(k-1)(k-2)/2$
- for all $0\leq m<i$ and $0<n<k$, a) holds
a) $ms+m(m-1)/2\neq is+i(i-1)/2-j-n(s+i+2j)-n(n-1)/2$
All variables are integers. I have tried computationally to find a counterexample but without luck.
This would imply the existence of several Self-avoiding walk on $\mathbb{Z}$ . (but not the converse.)