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asks how one might try to extend the Nekrasov-Okounkov formula by replacing the Plancherel measure on the Young lattice $\Bbb{Y}$ with another ergodic, central measure. In this discussion I want to instead replace the Young lattice $\Bbb{Y}$ by the Young-Fibonacci lattice $\Bbb{YF}$ which has a well-defined Plancherel measure in virtue of being a $1$-differential poset. Allow me to briefly review some basics of the Young-Fibonacci lattice before I state the putative $\Bbb{YF}$-version of the Nekrasov-Okounkov partition function.

Young-Fibonacci Preliminaries: Recall that a fibonacci word $u$ is a word formed out of the alphabet $\{1,2\}$. As a set $\Bbb{YF}$ is the collection of a (finite) fibonacci words and $\Bbb{YF}_n$ will denote the set of fibonacci words $u \in \Bbb{YF}$ of length $|u|=n$ where $|u|:= a_1 + \cdots + a_k$ and where $u=a_k \cdots a_1$ is the parsing of $u$ into its digits $a_1, \dots, a_k \in \{1,2 \}$. The adjective fibonacci reflects the fact that the cardinality of $\Bbb{YF}_n$ is the $n$-th fibonacci number. I will skip defining the poset structure on $\Bbb{YF}$ and instead I point the readers to the Wikipedia page https://en.wikipedia.org/wiki/Young–Fibonacci_lattice. Suffice it to say that when endowed with the appropriate partial order $\unlhd$ the set $\Bbb{YF}$ becomes a ranked, $1$-differential lattice. R. Stanley's concept of $1$-differential property (see https://en.wikipedia.org/wiki/Differential_poset) is key here because it implies that the function $\mu^{(n)}_\mathrm{P}: \Bbb{YF}_n \longrightarrow \Bbb{R}_{>0}$ defined by

restrict to a probability distribution on the set of covering relations $u \lhd v$ (i.e. edges in the Hasse diagram of $\Bbb{YF}$) for any fixed $u \in \Bbb{YF}_n$. We refer to $\mu^{(n)}_\mathrm{P}$ as the Plancherel measure for $\Bbb{YF}_n$. If $S:\Bbb{YF} \longrightarrow \Bbb{R}_{\geq 0}$ is some statistic let $\langle \S \rangle_n$ denote its expectation value with respect to the Plancherel measure, i.e.

We may visualize a fibonacci word $u \in \Bbb{YF}$ using a profile of boxes akin to the way depict a partition by its Young diagram; The following example with $u = 12112211$ should illustrate the concept of a Young-Fibonacci diagram clearly. For emphasis each digit of the fibonacci word $u$ is written directly underneath the corresponding column of boxes:

A fibonacci word $u$ will be synonymous with its Young-Fibonacci diagram and $\Box \in u$ will indicate membership of a box. The hook length $\mathrm{h}(\Box)$ of a box $\Box \in u$ is defined to be $1$ whenever it is the top row; otherwise $\mathrm{h}(\Box)$ equals $1$ plus the total number of boxes directly above it and to its left. For example the hook lengths of the boxes of $u = 12112211$ are indicated in the tableaux below:

asks how one might try to extend the Nekrasov-Okounkov formula by replacing the Plancherel measure on the Young lattice $\Bbb{Y}$ with another ergodic, central measure. In this discussion I want to instead replace the Young lattice $\Bbb{Y}$ by the Young-Fibonacci lattice $\Bbb{YF}$ which comes equipped with a Plancherel measure in virtue of being a $1$-differential poset. Allow me to briefly review some basics of the Young-Fibonacci lattice before I state the putative $\Bbb{YF}$-version of the Nekrasov-Okounkov partition function.

Young-Fibonacci Preliminaries: Recall that a fibonacci word $u$ is a word formed out of the alphabet $\{1,2\}$. As a set $\Bbb{YF}$ is the collection of a (finite) fibonacci words and $\Bbb{YF}_n$ will denote the set of fibonacci words $u \in \Bbb{YF}$ of length $|u|=n$ where $|u|:= a_1 + \cdots + a_k$ and where $u=a_k \cdots a_1$ is the parsing of $u$ into its digits $a_1, \dots, a_k \in \{1,2 \}$. The adjective fibonacci reflects the fact that the cardinality of $\Bbb{YF}_n$ is the $n$-th fibonacci number. I will skip defining the poset structure on $\Bbb{YF}$ and instead I point the readers to the Wikipedia page https://en.wikipedia.org/wiki/Young–Fibonacci_lattice. Suffice it to say that when endowed with an appropriate partial order $\unlhd$ the set $\Bbb{YF}$ becomes a ranked, modular (but not distributive), $1$-differential lattice. R. Stanley's concept of $1$-differential property (see https://en.wikipedia.org/wiki/Differential_poset) is key here because it implies that the function $\mu^{(n)}_\mathrm{P}: \Bbb{YF}_n \longrightarrow \Bbb{R}_{>0}$ defined by

restrict to a probability distribution on the set of covering relations $u \lhd v$ (i.e. edges in the Hasse diagram of $\Bbb{YF}$) for any fixed $u \in \Bbb{YF}_n$. We refer to $\mu^{(n)}_\mathrm{P}$ as the Plancherel measure for $\Bbb{YF}_n$. If $S:\Bbb{YF} \longrightarrow \Bbb{R}_{\geq 0}$ is some statistic let $\langle S \rangle_n$ denote its expectation value with respect to the Plancherel measure, i.e.

We may visualize a fibonacci word $u \in \Bbb{YF}$ using a profile of boxes akin to the way one depicts a partition by its Young diagram. The following example with $u = 12112211$ should illustrate the concept of a Young-Fibonacci diagram clearly. For emphasis each digit of the fibonacci word $u$ is written directly underneath the corresponding column of boxes:

A fibonacci word $u$ will be synonymous with its Young-Fibonacci diagram and $\Box \in u$ will indicate membership of a box. The hook length $\mathrm{h}(\Box)$ of a box $\Box \in u$ is defined to be $1$ whenever it is in the top row; otherwise $\mathrm{h}(\Box)$ equals $1$ plus the total number of boxes directly above it and to its right. For example the hook lengths of the boxes of $u = 12112211$ are indicated in the tableaux below:

restrict to a probability distribution on the set of covering relations $u \lhd v$ (i.e. edges in the Hasse diagram of $\Bbb{YF}$) for any fixed $u \in \Bbb{YF}_n$. The measure $\mu^{(n)}_\mathrm{P}$ is referred to as the Plancherel measure for $\Bbb{YF}_n$. If $S:\Bbb{YF} \longrightarrow \Bbb{R}_{\geq 0}$ is some statistic let $\langle \S \rangle_n$ denote its expectation value with respect to the Plancherel measure, i.e.

A fibonacci word $u$ will be synonymous with its Young-Fibonacci diagram and $\Box \in u$ will indicate membership of a box. The hook length $\mathrm{h}(\Box)$ of a box $\Box \in u$ is defined to be $1$ whenever it is the top row; otherwise $\mathrm{h}(\Box)$ equals $1$ plus the total number of boxes directly above it and to its right. For example the hook lengths of the boxes of $u = 12112211$ are indicated in the tableaux below:

restrict to a probability distribution on the set of covering relations $u \lhd v$ (i.e. edges in the Hasse diagram of $\Bbb{YF}$) for any fixed $u \in \Bbb{YF}_n$. We refer to $\mu^{(n)}_\mathrm{P}$ as the Plancherel measure for $\Bbb{YF}_n$. If $S:\Bbb{YF} \longrightarrow \Bbb{R}_{\geq 0}$ is some statistic let $\langle \S \rangle_n$ denote its expectation value with respect to the Plancherel measure, i.e.

A fibonacci word $u$ will be synonymous with its Young-Fibonacci diagram and $\Box \in u$ will indicate membership of a box. The hook length $\mathrm{h}(\Box)$ of a box $\Box \in u$ is defined to be $1$ whenever it is the top row; otherwise $\mathrm{h}(\Box)$ equals $1$ plus the total number of boxes directly above it and to its left. For example the hook lengths of the boxes of $u = 12112211$ are indicated in the tableaux below:

\begin{equation} F^\vee(z;t) \ = \ \sum_{k \geq 0} \, (-t)^{n-k} \, \sum_{n \geq 0} \, {z^n \over {n!}} \, \langle E_k \rangle_n \end{equation}

where $G_{\leq k}(z)$ is the generating function associated to the heap of inhomogeneous junk which, by induction, will have been previously evaluated. The homogeneous ODE has two nice independent solutions $Y_1(z) = e^z$ and $Y_2(z)= e^z \int {dz \over z} e^{-2z}$ whose Wronskian is just $W={z^{-1}}$. One starts the inductive engine beginning with $F^\vee_0(z) = e^z$.

\begin{equation} F^\vee(z;t) \ = \ \sum_{k \geq 0} \, (-t)^{n-k} \, \overbrace{\sum_{n \geq 0} \, {z^n \over {n!}} \, \langle E_k \rangle_n}^{F^\vee_k(z)} \end{equation}

where $G_{\leq k}(z)$ is the generating function associated to the heap of inhomogeneous junk which, by induction, will have been previously evaluated. The homogeneous ODE has two nice independent solutions $Y_1(z) = e^z$ and $Y_2(z)= e^z \int z^{-1} e^{-2z} dz$ whose Wronskian is just $W={z^{-1}}$. One starts the inductive engine beginning with $F^\vee_0(z) = e^z$.