Jump number and width

Abstract

The maximum jump number of ordered sets having width w and tower number t, denoted by s(w, t), satisfies

$$c_l tw\lg w \leqslant s\left( {w,t} \right) \leqslant tw\lg w$$

for some positive constants c ₁ and c ₂. Specifically, we can obtain c ₁=1/8 and c ₂<11/10. When w and t are sufficiently large and w is a power of 2, then

$$\left( {\frac{1}{2} - \varepsilon } \right)tw\lg w \leqslant s\left( {w,t} \right) < \frac{7}{{10}}tw\lg w$$

This gives an answer to a problem posed by W. T. Trotter ([3], Problem 15).