arXiv:1110.4379v2 [math.CO] 21 Oct 2011

Alexander Burstein’s Lo v ely Com binatorial Proof of John Noonan’s Beautiful Theorem

that the n um ber of n -p erm utations that co n tain the P attern 321 Exactly Once Equals

(3/n)(2n)!/((n-3)!( n+3)!)

Dor on ZEILBER GER

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Alex Bu rstein[1] ga ve a lo v ely com b inatorial pr o of of John No on an’s[2] lov ely theorem that the

n umber of n -p erm u tations that cont ain the pattern 321 exactly once equals

3

n



2 n

n +3



. Burstein’s

pro of can b e made ev en s horter as follo ws. L et C

n

:= (2 n )! / ( n !( n + 1)!) b e the Cata lan num b ers.

It is well -kno wn (and easy to see) that C

n

=

P

n − 1

i =0

C

i

C

n − 1 − i

. It is also well-kno wn (and f airly

easy t o see) that the num b er o f 321 -a v oidin g n -p er m utations e quals C

n

.

An y n -p erm utation, π , with exactly one 321 pattern can b e written as π

1

cπ

2

bπ

3

aπ

4

, where cba is

the uniqu e 321 p attern (so, of course a < b < c ). All the en tries to the left of b , except c , m ust

b e smaller than b , an d all t he en tr ies to the rig ht of b , except for a , m ust b e larger than b , or else

another 32 1 pattern w ould emerge. Hence σ

1

:= π

1

bπ

2

a is a 321-a v oidin g p erm utation of { 1 , . . . , b }

that do es n ot end with b and σ

2

:= cπ

3

bπ

4

is a 321-a vo iding p erm utation of { b, . . . , n } that d o es

not start with b . Th is is a bijection betw een the N o onan set a nd the set of pairs ( σ

1

, σ

2

) as ab ov e

(for some 2 ≤ b ≤ n − 1). F or any b the n umb er of p ossible σ

1

is C

b

− C

b − 1

. Similarly , the n umber

of p ossible σ

2

is C

n − b +1

− C

n − b

. Hence t he desired n umb er is

n − 1

X

b =2

( C

b

− C

b − 1

)( C

n − b +1

− C

n − b

) =

n

X

b =1

( C

b

− C

b − 1

)( C

n − b +1

− C

n − b

)

=

n

X

b =1

C

b

C

n − b +1

−

n

X

b =1

C

b

C

n − b

−

n

X

b =1

C

b − 1

C

n − b +1

+

n

X

b =1

C

b − 1

C

n − b

= C

n +2

− 2 C

n +1

− 2( C

n +1

− C

n

) + C

n

= C

n +2

− 4 C

n +1

+ 3 C

n

=

3

n



2 n

n + 3



.

References

1. Alex Burstein, A shor t pr o of for the numb e r of p ermutations c ontaining the p attern 321 e xactly

onc e , Elec. J. Com b. 18(2) (2011), #P21.

2. John No onan , The numb er of p ermutations c ontaining exactly one incr e asing subse qu enc e of

length thr e e , Discrete Ma th. 152 (1996), 307-31 3 .

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Departmen t of Ma thematics, Rutgers Universit y (New Brunswic k), H ill Center-Busc h Campus, 110 F relingh u ysen

Rd., Piscataw ay , NJ 08854 -8019 , USA. zeilb erg at math dot rut gers dot edu ,

http:/ /www. math.r utgers.edu/~zeilberg/ . Based on a mathematics colloquium talk at Howard Universit y , Oct.

14, 2011, 4:10-5 :00pm, where Alex Burstein w as presen ted with a $25 c hec k prize promise d in John No o nan’s 1996

Discrete Math pap er. Oct 18., 20 11. E xclusiv ely published in the Per sonal Journa l of Shalosh B. E khad and Doron

Zeilberger http:/ /www. math.r utgers.edu/~zeilberg/pj.html and arxiv.o rg . Supp orted in part by the NSF.

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