## 413_Unimodality of binomial coef & Sperner's theorem

Unimodality of Binomial Coefficients and Sperner’s Theorem If we examine the binomial coefficients in a row of Pascal’s triangle, we notice that the numbers increase for a while and then decrease. A sequence of numbers with this property is called unimodal. Theorem Let $n$ be a positive integer. The sequence of binomial coefficients $$ {{n} \choose {0}}, {{n} \choose {1}}, \ldots, {{n} \choose {n}} $$ is a unimodal sequence. Mor precisely, $$ {{n} \choose {0}} < {{n} \choose {1}} < \ldots < {{n} \choose {\lfloor n/2\rfloor}} $$ and $$ {{n} \choose {\lceil n/2 \rceil}} > \ldots {{n} \choose {n-1}} > {{n} \choose {n}} $$...