## n-series

For discussing Olympiad Level Combinatorics problems
Thamim Zahin
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

### n-series

Given any integer \$n\geq 3\$. A finite series is called \$n\$-series if it satisfies the following two conditions
\$1)\$ It has at least \$3\$ terms and each term of it belongs to \$\{ 1,2,...,n\}\$
\$2)\$ If series has \$m\$ terms \$a_1,a_2,...,a_m\$ then \$(a_{k+1}-a_k)(a_{k+2}-a_k)<0\$ for all \$k=1,2,...,m-2\$

How many \$n\$-series are there \$?\$
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SN.Pushpita
Posts: 11
Joined: Thu Aug 10, 2017 1:44 pm

### Re: n-series

First observe that for any sequence, if the first 3 terms are a,b and c respectively,then either b<a <c or c <a <b will hold. Now repeating the condition given in the question and fixing the number of terms u can get the order of each sequence.So FOR each subset of cardinality >=3 of {1,2,3,....n} u can get 2 n-series.So rest is easy.