hw6 - Math 146: Algebraic Combinatorics Homework 6 This...

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Math 146: Algebraic Combinatorics Homework 6 This problem set is due Friday, May 7. Do problems 3.2 and 3.5, in addition to the following: GK6.1. Theorem 3.16.2 in the book generalizes the equality between odd partitions and distinct partitions to partitions not divisible by b = q + 1 and partitions in which each part appears less than b times. For instance, when b = 3, we learn that ( 1 + x + x 2 )( 1 + x 2 + x 4 )( 1 + x 3 + x 6 ) ··· = 1 ( 1 - x )( 1 - x 2 )( 1 - x 4 )( 1 - x 5 )( 1 - x 7 ) ··· using the following identities: 1 1 - x = ( 1 + x + x 2 )( 1 + x 3 + x 6 )( 1 + x 9 + x 18 ) ··· 1 1 - x 2 = ( 1 + x 2 + x 4 )( 1 + x 6 + x 12 )( 1 + x 18 + x 36 ) ··· 1 1 - x 4 = ( 1 + x 4 + x 8 )( 1 + x 12 + x 24 )( 1 + x 36 + x 72 ) ··· . . . As I did in class when b = 2, use these non-cancelling identities to obtain an explicit bijection between partitions not divisible by b and partitions whose parts occur less than b times. GK6.2.
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This note was uploaded on 05/19/2010 for the course MATH mat 146 taught by Professor Gregkuperberg during the Spring '10 term at UC Davis.

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hw6 - Math 146: Algebraic Combinatorics Homework 6 This...

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