co.combinatorics – Young-Fibonacci version of Nekrasov-Okounkov

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 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

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 $F^vee_0(z) = e^z$.

Question: Has the linear recurrence satisfied by $sigma_k(n)$
or else the hierarchy of 2nd order inhomogeneous ODEs been
studied ?

thanks, ines.