1.Let S ={1 , 2 ,...,n }. Apermutation a 1 a 2 ...a n of S issaidtobe alternating if a 1 a 3 a 5

1.LetS={1,2,...,n}.

Apermutationa₁a₂...a_nofSissaidtobealternatingif

a₁₂>a₃₄>a₅<.

Example:

n	AlternatingPermutations	No.ofsuchpermutations
1	1	1
2	12	1
3	132,231	2
4	1324,1423,2314,2413,3412	5
5	13254,14253,14352,15243,	16
	15342,23154,24153,24351,
	25143,25341,34152,34251,
	35142,35241,45132,45231

Apermutationa₁a₂...a_nofSissaidtobereversealternatingif

a₁>a₂₃>a₄₅>.

Example:

n	ReverseAlternatingPermutations	No.ofsuchpermutations
1	1	1
2	21	1
3	312,213	2
4	4231,4132,3241,3142,2143	5
5	53412,52413,52314,51423,	16
	51324,43512,42513,42315,
	41523,41325,32514,32415,
	31524,31425,21534,21435

UsethePigeonholePrincipletoshowthatamong97pointsinaregularhexagonofsidelength4cm,thereare2whosedistanceisatmost1cmapart.