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

Unread post by Thamim Zahin » Sun Oct 30, 2016 7:55 pm

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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Joined:Thu Aug 10, 2017 1:44 pm

Re: n-series

Unread post by SN.Pushpita » Thu Feb 15, 2018 2:24 pm

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.

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