DATE:
Tuesday, October 11 (2011), at 1:55pm
PLACE: LIT 368
SPEAKER:
Alexander Berkovich
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TITLE:
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Sign Reversing Involutions. Three pedagogical examples. (Part 2)
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ABSTRACT:
This is a pedagogical introduction to sign reversing involution technique.
I will illustrate the power of this technique by establishing three formulas below.
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q-binomial identity
$$
\sum_{k=0}^n (-1)^k q^{k(k-n)+n(n+1)/2}
\left[\begin{matrix} n \\ k \end{matrix}\right]_q^2
= (-1)^{n/2} q^{ n(n+2)/4 }
\left[\begin{matrix} n \\ n/2 \end{matrix}\right]_{q^2}
$$
if n is even, and
$$
\sum_{k=0}^n (-1)^k q^{k(k-n)+n(n+1)/2}
\left[\begin{matrix} n \\ k \end{matrix}\right]_q^2
=0,
$$
if n is odd.
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Euler's Pentagonal Number Theorem:
$$
\prod_{n=1}^\infty (1 - q^n) = \sum_{k=-\infty}^\infty (-1)^k q^{k(3k+1)/2}.
$$
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Andrews-Garvan (1988) identity relating vector partitions with prescribed cranks
with ordinary partitions with prescribed cranks.
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