This question addresses a hierarchy of linear recurrences

which arise from an attempt to generalize the Nekrasov-Okounkov

formula to the Young-Fibonacci setting.

A related posting

extensions of the Nekrasov-Okounkov formula

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 its own *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 $1$-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

begin{equation}

begin{array}{ll}

mu^{(n)}_mathrm{P}(u)

&displaystyle := {1 over {n!}} , dim^2(u) quad text{where} \

dim(u)

&displaystyle :=

# left{

begin{array}{l}

text{all saturated chains $(u_0 lhd cdots lhd u_n)$ in $Bbb{YF}$} \

text{starting with $u_0 = emptyset$ and ending at $u_n =u$}

end{array}

right}

end{array}

end{equation}

is a strictly positive probability distribution

on $Bbb{YF}_n$ for each $n geq 0$. Furthermore these distributions are

*coherent* in the sense that the ratios

begin{equation}

tilde{mu}_mathrm{P}(u lhd v) :=

{mu^{(n+1)}_mathrm{P}(v) over {mu^{(n)}_mathrm{P}(u)}}

end{equation}

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.

begin{equation}

langle S rangle_n := sum_{|u|=n} , {dim^2(u) over {n!}} , S(u)

end{equation}

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:

begin{equation}

begin{array}{cccccccc}

& Box & & & Box & Box & & \

Box & Box & Box & Box & Box & Box & Box & Box \

1 & 2 & 1 & 1 & 2 & 2 & 1 & 1

end{array}

end{equation}

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:

begin{equation}

begin{array}{cccccccc}

& boxed{1 } & & & boxed{1 } & boxed{1 } & & \

boxed{11} & boxed{10} & boxed{8 } & boxed{7 }

& boxed{6 } & boxed{4 } & boxed{2 } & boxed{1 }

end{array}

end{equation}

These graphical conventions allows us to reformulate

the value of the Plancherel measure as a product of

hook-lenghts, i.e.

begin{equation}

mu^{(n)}_mathrm{P}(u) = prod_{Box , in , u} , {n! over

{mathrm{h}^2(Box)} }

end{equation}

This is a non-trivial observation made by R. Stanley in the course

of his work examining differential posets.

**The $Bbb{YF}$-version of the Nekrasov-Okounkov partition function:**

For a fibonacci words $u in Bbb{YF}$

define a $t$-statistic

$H_t(u) := prod_{Box , in , u} , big(mathrm{h}^2(Box) – t big)$ and the *$Bbb{YF}$-Nekrasov-Okounkov* partition function as

begin{equation}

begin{array}{ll}

F(z;t)

&displaystyle = sum_{n geq 0} {z^n over {n!}}

, langle H_t rangle_n \

&displaystyle = sum_{n geq 0} {z^n over {n!}} ,

sum_{|u|=n} , {dim^2(u) over {n!}} , H_t(u)

end{array}

end{equation}

It will be convenient, when dealing with expansions into elementary

symmetric polynomials, to make the change of variable $z mapsto -z$

and consider $F^vee(z;t)

:= F(-z;t)$ instead; the effect of this sign-change is to

replace the statistic $H_t(u)$ by $H^vee_t(u) := prod_{Box , in , u} , big(t -mathrm{h}^2(Box) big)$ in the definition of the partition

function. After expanding into elementary symmetric polynomial $E_k$ we

get

begin{equation}

H^vee_t(u) = sum_{k=1}^n , (-t)^{n-k} , E_k big( mathrm{h}^2(Box) big)_{Box , in , u}

end{equation}

and

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}

which effectively reduces the problem of calculating $F^vee(z;t)$

to the problem of evaluating the expectation values

$langle E_k rangle_n$.

**Evaluating expectation values:**

Fibonacci words $u in Bbb{YF}_n$ with $n geq 2$ can be separated into two

disjoint groups: Those of the form $u=1v$ for $v in Bbb{YF}_{n-1}$

and those of the form $u=2v$ for $v in Bbb{YF}_{n-2}$. Depending on

whether the prefix of $u$ is $1$ or $2$ we can write down a recursive

formula for the value of $E_k(u) := E_k big( mathrm{h}^2(Box) big)_{Box , in , u}$ by analyzing the hook length(s) of the box(es) in the left-most

column, specifically:

begin{equation}

begin{array}{lll}

E_k(1v)

&= E_k(v) + n^2E_{k-1}(v)

&text{if} |v| = n-1 \

E_k(2v)

&= E_k(v) + (n^2+1)E_{k-1}(v) + n^2E_{k-2}(v)

&text{if} |v| = n-2

end{array}

end{equation}

Using the observation that $dim(1v) = dim(v)$ and

$dim(2v) = (|v| + 1)^2 dim(v)$ we may conclude

begin{equation}

langle E_k rangle_n

= left{

begin{array}{l}

displaystyle {1 over n} langle E_k rangle_{n-1}

+ {n-1 over n} langle E_k rangle_{n-2} \ \

displaystyle + n langle E_{k-1} rangle_{n-1} +

{(n-1)(n^2+1) over n} langle E_{k-1} rangle_{n-2} +

n(n-1) langle E_{k-2} rangle_{n-2}

end{array}

right.

end{equation}

If we set $sigma_k(n) := {1 over {n!}} , langle E_k rangle_n$ then

the above recursion can be rewritten as:

begin{equation}

n^2sigma_k(n) =

underbrace{sigma_k(n-1) + sigma_k(n-2)}_{text{homogeneous part}}

+

underbrace{n^2sigma_{k-1}(n-1)

+(n^2 +1)sigma_{k-1}(n-2) + n^2sigma_{k-2}(n-2)}_{text{inductive heap of inhomogeneous junk}}

end{equation}

which can be converted, using the usual yoga of generating functions, into the following second order inhomogeneous ODE for $F^vee_k(z) :=

sum_{n geq 0} sigma_k(n) z^n$

begin{equation}

z^2 , {d^2 over {dz^2}} , F^vee_k +

z , {d over {dz}} , F^vee_k +

big(z^2 + z big) , F^vee_k

=

G_{leq k}(z) + big( sigma_k(1) – sigma_k(0) big)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 $F^vee_0(z) = e^z$. For $k=1$ its not hard to see that

$sigma_1(1)=1$ and $sigma_1(0)=0$ while

begin{equation}

begin{array}{ll}

displaystyle G_{leq 1}(z)

&displaystyle = {z^2 over {1-z}} +

2 sum_{n geq 2} , n^2 z^n \

&displaystyle = {z^2 over {1-z}} +

2z , Bigg( {1+z over {(1-z)^3}} , – , 1 Bigg)

end{array}

end{equation}

so the ODE for $F^vee_1(z)$ becomes

begin{equation}

z^2 , {d^2 over {dz^2}} , F^vee_1 +

z , {d over {dz}} , F^vee_1 +

big(z^2 + z big) , F^vee_1

=

z + G_{leq 1}(z)

end{equation}

By variation of parameters, a particular inhomogeneous solution is

begin{equation}

begin{array}{rl}

displaystyle Y_mathrm{particular}(z)

&displaystyle = V_1(z) cdot e^z + V_2(z) cdot

e^z overbrace{int {dz over z} , e^{-2z} }^{gamma(z)} \

displaystyle V_1(z)

&displaystyle = -int z , e^z , gamma(z) ,

Big(z + G_{leq 1}(z) Big) , dz \

displaystyle V_2(z)

&displaystyle = int z , e^z ,

Big(z + G_{leq 1}(z) Big) , dz

end{array}

end{equation}

**Question:** Has the linear recurrence satisfied by $sigma_k(n)$

or else the hierarchy of 2nd order inhomogeneous ODEs been

studied ?

thanks, ines.