A simple inequality

santi_spadaro at virgilio.it santi_spadaro at virgilio.it
Tue Mar 30 18:20:07 CEST 2004

There's an old Monthly article (Vol. 87, n.9) of Solomon Golomb which deals
with iterated binomial coefficients. Among the many elegant results which
are proved I came to this simple inequality:

Binomial (Binomial (n,2), 3) > Binomial (Binomial (n,3), 2)

While providing both an algebraic and combinatorial proof for most of the
other identities, here he only gives a proof based on numerical estimates
of the Binomial coefficients. This result also follows trivially from two
of the reduction formulas which Golomb states at the end of the article,

Binomial (Binomial (n,2),3)= Binomial (n+1,6)+13*Binomial (n+2, 6)+Binomial

Binomial (Binomial (n,3),2) = 6*Binomial (n+2,6)+3*Binomial (n+1,6)+Binomial

I would be very pleased to see a combinatorial proof of the above fact,
say an injection between 3-multigraphs on 2 edges and n vertices and graphs
on 3 edges and n vertices. Can you think of any?


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