From e749eb870889cdba2e0f7cb2c3abeb0cb5660632 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Wed, 20 Apr 2022 11:03:16 -0400 Subject: [PATCH] Started cleaning Appendix B. --- app_hard_lem-mult-p.tex | 4 ++-- main.pdf | Bin 1010536 -> 1010410 bytes main.synctex.gz | Bin 811566 -> 811382 bytes 3 files changed, 2 insertions(+), 2 deletions(-) diff --git a/app_hard_lem-mult-p.tex b/app_hard_lem-mult-p.tex index 0b4528e..2df39ce 100644 --- a/app_hard_lem-mult-p.tex +++ b/app_hard_lem-mult-p.tex @@ -13,8 +13,8 @@ Given that we then have $2\kElem + 1$ distinct values of $\rpoly_{G}^\kElem(\pro Next, we show why we can compute $\numocc{G}{\kmatch}$ from $c_{2k}$ in $O(1)$ additional time. We claim that $c_{2\kElem}$ is $\kElem! \cdot \numocc{G}{\kmatch}$. This can be seen intuitively by looking at the expansion of the original factorized representation \[\poly_{G}^\kElem(\vct{X}) = \sum_{\substack{(i_1, j_1),\cdots,(i_\kElem, j_\kElem) \in E}}X_{i_1}X_{j_1}\cdots X_{i_\kElem}X_{j_\kElem},\] -where a unique $\kElem$-matching in the multi-set of product terms can be selected $\prod_{i = 1}^\kElem i = \kElem!$ times. -Indeed, note that each $\kElem$-matching $(i_1, j_1)\ldots$ $(i_k, j_k)$ in $G$ corresponds to the monomial $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ in $\poly_{G}^\kElem(\vct{X})$, with distinct indexes, and this implies that each distinct $\kElem$-matching appears the exact number of permutations that exist for its particular set of $\kElem$ edges, or $k!$. +where a unique $\kElem$-matching in the multi-set of product terms has $\kElem$ distinct $\inparen{i_\ell, j_\ell}$ index pairs. Further, any monomial composed of such a distinct set of $2\kElem$ variables will be produced $\kElem!$ times in a $\kElem$-wise product of the sum of a set of elements with itself. This is true because each (identical) product term contains each of the $\kElem$ distinct elements, giving us $\kElem\cdot\kElem-1\cdots 1 = \kElem!$ permutations of a distinct $\kElem$-matching. %Such must be the case since each product term has since this is the number of permutations for a given monomial.can be selected $\prod_{i = 1}^\kElem i = \kElem!$ times. +%Indeed, note that each $\kElem$-matching $(i_1, j_1)\ldots$ $(i_k, j_k)$ in $G$ corresponds to the monomial $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ in $\poly_{G}^\kElem(\vct{X})$, with distinct indexes, and this implies that each distinct $\kElem$-matching appears the exact number of permutations that exist for its particular set of $\kElem$ edges, or $k!$. Since, as noted earlier, $c_{2\kElem}$ represents the number of monomials with $2\kElem$ distinct variables, then it must be that $c_{2\kElem}$ is the overall number of $\kElem$-matchings. 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