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## co.combinatorics – extensions of the Nekrasov-Okounkov system

In what follows $lambda$ will breathe an integer partition (of dimension $|lambda| = n$),

the notation $b in lambda$ will explain that $b$ is a cell or *field* within the Young diagram of $lambda$, and $mathrm{h}(b)$ will denote the

*swipe size* of the swipe clear by $b$ in $lambda$‘s Young diagram.

Furthermore $dim(lambda)$ will denote the variety of saturated chains

$lambda^{(0)} subset cdots subset lambda^{(n)}$ within the Young lattice $Bbb{Y}$ birth on the vacant partition $lambda^{(0)} = emptyset$ and

ending at $lambda^{(n)} = lambda$. Equivalently $dim(lambda)$ is the dimension of the corresponding irreducible illustration $V_lambda$ of the symmetric group $S_n$.

The Nekrasov-Okounkov system is the next $t$-deformation of the classical id for the partition duty, particularly:

commence{equation}

sum_{n geq 0} , {z^n over {n !}} , sum_lambda ,

{dim^2(lambda) over {n!}} , prod_{b in lambda} , huge(

mathrm{h}^2(b) – t huge) =

prod_{okay geq 1} , huge(1 – z^okay huge)^{t-1}

aim{equation}

with the specialization $t=0$ equivalent to the well-known id for the producing duty of $p(n)$, the variety of integer partitions of $n geq 0$:

commence{equation}

sum_{n geq 0} , p(n) , z^n = prod_{okay geq 1} , huge( 1 – z^okay huge)^{-1}

aim{equation}

My first query issues the interior sum

commence{equation}

sum_lambda ,

{dim^2(lambda) over {n!}} , prod_{b in lambda} , huge(

mathrm{h}^2(b) – t huge)

aim{equation}

which we will perceive as an expectation worth for the $t$-statistic $H_t(lambda) = prod_{b in lambda} , huge(mathrm{h}^2(b) – t huge) $ with respect to the

*Plancherel* touchstone $mu^{(n)}_mathrm{P}(lambda) = {1 over {n!}} dim^2(lambda)$ on the clique of partitions of dimension $|lambda| =n$. Let’s denote this expectation worth as $langle H_t rangle_{mathrm{P},n}$. Clearly it’s a polynomial in $t$.

**Question 1:** Is there a pleasant (e.g. combinatorial) system

describing how $langle H_t rangle_{mathrm{P},n}$ factorizes

as a polynomial in $t$?

Instead of utilizing the (household of) Plancherel touchstone(s) $mu^{(n)}_mathrm{P}$

we might employ a coherent, ergodic household of measures $mu^{(n)}_varphi$

related to a different altenative of normalised, minimal, harmonic duty $varphi: Bbb{Y} longrightarrow Bbb{R}_{>0}$. Recall that the

touchstone $mu_varphi^{(n)}$

on the clique of partitions $lambda$ of dimension $|lambda| = n$ is outlined

by

commence{equation}

mu_varphi^{(n)} (lambda) := dim(lambda) , varphi(lambda)

aim{equation}

Among the provision of minimal, normalised, harmonic capabilities

are the *Schur*-states $varphi_{bf x}$ outlined by $varphi_{bf x}(lambda) = {scriptstyle frak{S}}_lambda(x_1, x_2, x_3, dots)$ the place

${scriptstyle frak{S}}_lambda$ is the Schur duty related to

$lambda in Bbb{Y}$ and

the place ${bf x} = (x_1, x_2, x_3, dots)$ is any sequence of constructive actual numbers with $0 leq x_k leq 1$

and $x_{okay+1} leq x_k$ for all $okay geq 1$ and with convergent sum

$sum_{okay geq 1} x_k = 1$. Accordingly outline the corresponding *Schur* touchstone(s) by

$mu^{(n)}_{bf x}(lambda) = dim(lambda) , {scriptstyle frak{S}}_lambda(x_1, x_2, x_3, dots)$.

**Question 2:** Is a model of the Nekrasov-Okounkov system

identified the place the position of the Plancherel touchstone $mu^{(n)}_mathrm{P}$ is changed by a Schur touchstone $mu^{(n)}_{bf x}$? Specifically

an id for

commence{equation}

sum_{n geq 0} , {z^n over {n !}} , sum_lambda ,

mu^{(n)}_mathrm{x} , prod_{b in lambda} , huge(

mathrm{h}^2(b) – t huge) = ?

aim{equation}

Thanks, ines.

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