# 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.