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These are my notes for the

Kim Zeng involution

The Linearization formula

{$\mathcal{L}(P_n(x)^2)=n!(c)_n(\alpha\beta)^n$} where {$(c)_n$} is the rising factorial

Combinatorially, Kim and Zeng's formula for linearization coefficients yields

{$\mathcal{L}(P_n(x)^2)=(\alpha\beta)^n\sum_{\sigma\in\mathcal{D}_{(n,n)}}c^{\textrm{cyc}\;\sigma}$}

{$\sum_{\sigma\in\mathcal{D}_{(n,n)}}c^{\textrm{cyc}\;\sigma}=n!(c)_n$}

For example, if we define our generalized derangements on the odds {$\{1,3,5,7,\dots\}$} and the evens {$\{a,b,c,d,\dots\}$}, and consider all permutations of the evens, we have:

 {$c$} {$(1 a)$} {$2c(c+1) = 2(c^2 + c)$} {$(1 a)(3 b),(1 a 3 b)$} {$6c(c+1)(c+2) = 6(c^3 + 3c^2 + 2c)$} {$(1 a)(3 b)(5 c), (1 a 3 b)(5 c), (1 a 5 b)(3 c), (1 a)(3 b 5 c), (1 a 3 b 5 c), (1 a 5 b 3 c)$} {$24c(c+1)(c+2)(c+3) = 24(c^4 + 6c^3 + 11c^2 + 6c)$} {$(1 a)(3 b)(5 c)(7 d),$} {$(1 a 3 b)(5 c)(7 d), (1 a 5 b)(3 c)(7 d), (1 a 7 b)(3 c)(5 d), (1 a)(3 b 5 c)(7 d), (1 a)(3b 7c)(5 d), (1 a)(3 b)(5 c 7 d),$} {$(1 a 3 b 5 c)(7 d), (1 a 5 b 3 c)(7 d), (1 a 3 b 7 c)(5 d), (1 a 7 b 3 c)(5 d), (1 a 5 b c 7)(d 3),(1 a 7 b 5 c)(3 d), (1 a)(3 b 5 c 7 d), (1 a)(3 b 7 c 5 d)$} {$(1 a 3 b)(5 c 7 d), (1 a 5 b)(3 c 7 d), (1 a 7 b)(3 c 5 d),$} {$(1 a 3 b 5 c 7 d), (1 a 3 b 7 c 5 d), (1 a 5 b 3 c 7 d), (1 a 5 b 7 c 3 d), (1 a 7 b 3 c 5 d), (1 a 7 b 5 c 3 d)$}

If {$c=1$}, then we get {$n!n!$} terms. In other words, {$(1)_n=n!$}

Moments

The formula can be interpreted by calculating the moments {$\mathcal{L}(x^n)$}.

{$\mathcal{L}(x^n)=\sum_{\sigma\in S_n} (-\alpha)^{\textrm{asc}\;\sigma} (-\beta)^{\textrm{desc}\;\sigma} f^{\textrm{fix}\;\sigma} c^{\textrm{cyc}\;\sigma}$}

For {$A_{(n,n)}$}, skew derangements and generalized derangements are the same

Any generalized derangement without color matches is a skew derangement. Note that in the case of {$A_{(n,n)}$} there are no double ascents (and no double descents) and so there are no color matches.

A skew derangement is a permutation {$\pi$} of {$A_{(n,n}$} that has no color matches, and for which if {$c(a)=c(\pi(a))$}, then {$c(\pi^{-1}(a))\neq c(a)$}, {$c(\pi(a))\neq c(\pi^2(a))$} and the smaller of {$\{a,\pi(a)\}$} is a valley and the larger is a peak. But this case supposes that there are three colors:

{$c(\pi^{-1}(a)) > c(a)=c(\pi(a)) > c(\pi^2(a))$} and {$a <\pi(a)$}

or

{$c(\pi^{-1}(a)) < c(a)=c(\pi(a)) < c(\pi^2(a))$} and {$a > \pi(a)$}

Consequently, since we have two colors, then we simply have that {$c(a)\neq c(\pi(a))$} for all {$a\in (n,n)$}, which means that the skew derangements of {$A_{(n,n)}$} are the generalized derangements of {$A_{(n,n)}$}.

Causal trees

L-graphs

This page was last changed on February 02, 2024, at 07:54 PM