Random Weighted Staircase Tableaux
Amanda Lohss and Pawel Hitczenko
Department of Mathematics
Drexel University
Definition
A staircase tableau of size n is a Young diagram of shape (n, n-1, ..., 1) such that:
1
2
3
4
The boxes are empty or contain an α, β, γ, or δ.
All boxes in the same column and above an α or γ are empty.
All boxes in the same row and to the left of an β or δ are empty.
Every box on the diagonal contains a symbol.
α/β-Staircase Tableaux
W.L.O.G. we can study α/β-staircase tableaux since α’s and γ’s follow the same rules and
β’s and δ’s follow the same rules.
If m := min{l ≥ 1 : jl ≤ jl+1 − k}, then
The generating function of α/β-staircase tableaux is:
X
Zn (α, β) :=
wt(S) = Zn (α, β, 0, 0) = αn β n (a + b)n
Pn (α1 , αjk2 , . . . , αjkr ) =
b
k−m−1
X
Ck,h
b
h=0
.
Ck,h := #ways you can have h D-connected symbols in a k × k tableaux
Example: An α/β-Staircase Tableau
Let Sn be the set of all staircase tableaux of size n. The weight of S ∈ Sn is the product of
all symbols in S:
wt(S) = αNα β Nβ γ Nγ δ Nδ .
P
It has been shown that the generating function Zn (α, β, γ, δ) := S∈Sn wt(S) is equal to the
product:
n−1
Y
Zn (α, β, γ, δ) =
(α + β + δ + γ + i(α + γ)(β + δ)).
Definition: D-connected
α
α
Define xji , i 6= 1 to be D-connected if its αβ-path is connected to an αjk .
α
β
β
Definition: αβ-path
α
β
β
i=0
α
Connections in Combinatorics and Analysis
β
The generating function for staircase tableaux has been used to give a formula for the
moments of Askey-Wilson polynomials.
Staircase tableaux have also inherited many interesting properties from other types of
tableaux as there are bijections between staircase tableaux of size n, permutation tableaux of
length n + 1, alternative tableaux of length n, and tree-like tableaux of length n + 2.
Application: The Asymetric Simple Exclusion Process (ASEP)
The ASEP can be defined as a Markov chain with n sites, with at most one particle
occupying each site. Particles may jump to any neighboring empty site with rate u to the
right and rate q to the left. Particles may enter and exit at the first site with rates α and γ
respectively. Similarly, particles may enter and exit the last site with rates δ and β.
The α/β-staircase tableau obtained from above by replacing γ’s with α’s and δ’s with β’s.
It’s weight is α5 β 5 .
Definition: Random α/β-Staircase Tableaux
For all n ≥ 1, α, β ∈ [0, ∞) with (α, β) 6= (0, 0), Sn,α,β is defined to be a random
α/β-staircase tableau in S n with respect to the probability distribution on S n given by:
For any xhi , i 6= 1, i 6= k, define its αβ-path to be the path defined recursively by the
following rules:
1 If ∃l, l ≤ i, such that β l (i.e. there is a β below in the same column), then connect those
h
boxes (n − i − h + 2, h) and (n − l − h + 2, h).
i+h−j
2 If ∃j, j ≥ i, such that α
(i.e. there is an α west in the same row), then connect
j
those boxes (n − i − h + 2, h) and (n − i − h + 2, j).
3 If αk and i + h = k + j (i.e. there is an alpha on the k th diagonal in the same row), then
j
connect those boxes (n − i − h + 2, h) and (n − k − j + 2, j)
Properties of D-connected Symbols:
1
2
Nα N β
∀S ∈ S n ,
P(Sn,α,β = S) =
wt(S)
α β
=
.
Zn (α, β) Zn (α, β)
The ASEP is an interesting particle model that has been studied extensively in mathematics
and physics. It has also been studied in many other fields, including computational biology,
and biochemistry, specifically as a primitive model for protein synthesis.
All of our results for random α/β-staircase tableaux can be extended to random staircase
tableaux with all four parameters, α, γ, β, δ by considering Sn,α+γ,β+δ and randomly replacing
γ
δ
and similarly, each β with δ with probability β+δ
each α with γ with probability α+γ
independently for each occurrence.
Staircase tableaux were introduced per a connection between the steady state distribution of
the ASEP and the generating function for staircase tableaux.
Hitczenko & Janson’s Previous Results & Conjecture
If An := the number of α’s then as n → ∞:
3
Any symbol in the same row or the same column as a D-connected symbol is also
D-connected.
b
There are at most kb − m − 1 D-connected symbols in S.
Each D-connected symbol can be paired uniquely with an opposite diagonal symbol.
Example: A Staircase Tableaux and its αβ-paths
The connection between staircase tableaux and the ASEP requires an extension of the
preceding definition which requires filling empty boxes with parameters u’s and q’s. Each
such staircase tableau is associated with a state of the ASEP by aligning the Markov chain
with the diagonal entries of the staircase tableau. A site is filled if the corresponding diagonal
entry is an α or a γ and a site is empty if the corresponding diagonal entry is a β or a δ.
Each staircase tableau’s associated state of the ASEP is called its type.
α
•α α
β
◦α
δ
β
γ
δ
β
γ
β
α q
q
γ
q
u
α
q
q
q
q
q
δ
•
u
u
β u
γ
q
q
q
u
u
β
q
γ
β
δ
•
1
d
An → Pois
2
1
d
Bn → Pois
2
Theorem: The Distribution of Boxes
Let 1 ≤ j1 < ... < jr ≤ n − k + 1. If
jl ≤ jl+1 − k, ∀l = 1, 2, ..., r − 1
•
◦
◦
◦
then
Pn,α,β (αjk1 , ..., αjkr )
r
Y
b + jr −l+1 − 2r + 2l − 1
1
=
+O
.
(n + a + b − 2r + 2l − 1)2
(n + a + b)r +1
l=1
Otherwise,
2 3 2 3
A staircase tableau of size 7 with weight α β δ γ and its extension to a staircase tableau of
weight α2 β 3 δ 2 γ 3 u 6 q 13 and type • • ◦ • ◦ ◦ ◦.
Pn,α,β (αj1 , ..., αjr ) = O
Amanda Lohss and Pawel Hitczenko
1
(n + a + b)r
α
β
Theorem: Asymptotic Distribution of the k th Diagonal
If An := the number of α’s along the k th main diagonal, then as n → ∞:
◦
α
β
Conjecture: The number of α’s and the number of β’s along the other diagonals is
asymptotically Poisson.
Example: A Staircase Tableau and its Extension
α
•α
•β
An − n/2 d
√
→ N(0, 1/12)
n
Using this association, it was shown that the steady state probability that the ASEP is in
state η is given by:
P
T ∈T wt(T )
,
Zn
where T is the set of all staircase tableau of type η.
•α α
β
◦α
•β
Connection to the ASEP
γ
bm
Pn−h−2m (αjkm+1 , . . . , αjkr ).
(n + a + b − 1)h+2m
where
S∈S n
Generating Function for Staircase Tableaux
α
Lemma for Induction
.
Random Weighted Staircase Tableaux
β
β
A staircase tableau of size 11. D is denoted by a black line. The αβ-paths are pictured with
blue lines. In this staircase tableau, there are five D-connected symbols (blue bullets).
Selected References
S. Corteel and L. K. Williams. Tableaux combinatorics for the asymmetric exclusion
process. Adv. Appl. Math., 39:293-310, 2007.
S. Corteel and L. K. Williams. Staircase tableaux, the asymmetric exclusion process, and
Askey-Wilson polynomials. Proc. Natl. Acad. Sci., 107(15):6676-6730, 2010.
S. Corteel and L. K. Williams. Tableaux combinatorics for the asymmetric exclusion
process and Askey-Wilson polynomials. Duke Math. J., 159:385-415, 2011.
S. Corteel, R. Stanley, D. Stanton, and L. Williams. Formulae for Askey-Wilson moments
and enumeration of staircase tableaux. Trans. Amer. Math. Soc., 364(11):6009-6037,
2012.
S. Dasse-Hartaut and P. Hitczenko. Greek letters in random staircase tableaux. Random
Struct. Algorithms, 42:73-96, 2013.
P. Hitczenko and S. Janson. Weighted random staircase tableaux.
Combin. Probab. Comput., 23 (2014), 1114–1147.
P. Hitczenko and A. Parshall. On the distribution of the parameters in weighted random
staircase tableaux. Discrete Mathematics and Theoretical Computer Science, proc. BA,
2014, 157–168 (also arXiv:1404.3446)