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

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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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