The discrete two-dimensional Toda equation gives nice formulae for reverse plane partitions
Shuhei Kamioka
Graduate School of Informatics, Kyoto University, Japan
Abstract: We clarify a close connection between reverse plane partitions(RPPs) and an integrable dynamical system called the discrete
two-dimensional (2D) Toda equation. We show that a nice partition function(with a product expression) can be obtained from each non-vanishing solution
to the discrete 2D Toda equation. As an example we derive a partition
function which generalizes MacMahon’s formula and Gansner’s hook formulafrom a specific solution to the discrete 2D Toda equation.
Remark: A similar result for rectangular-shaped plane partitions is discussed in[K1,K2] where biorthogonal polynomials are used to find a multiplicativepartition function.

RPPs π ∈ RPP(λ, c) are in one-to-one correspondence with c-tuples of
non-intersecting paths ( P1, . . . , Pc ) on a lattice graph determined from theYoung diagram λ.

Main Theorem
(s,t)
(s,t)
Let qn 6= 0, en 6= 0 solve the discrete 2D Toda equation (5). Let λ be aYoung diagram with a rows and b columns. Let c ≥ 0. Then

Example: For the RPP (1) of shape λ = (5, 4, 4, 2, 1)

π=002230 1 1 22 3 44 4 44

←
→1:1

P1

Each non-vanishing solution to the discrete 2D Toda eq. gives
a weight v for RPPs having a nice formula!

P2
P3

Definition

Example: An RPP π ∈ RPP(λ, 4) of shape λ = (5, 4, 4, 2, 1)

weakly increasing00223

λ1 = 5 cell(s)λ2 = 4 —λ3 = 4 —λ4 = 2 —λ5 = 1 —0 1 1 22 3 44 4 44We have the following solution to the discrete 2D Toda eq. (5):Weight by lattice paths

an RPP π ∈ RPP(λ, c) as follows. Let ( P1, . . . , Pc ) ←→ π by the one-to-one
correspondence. Thenc(4)k=1 e∈ Pkk =1

1
11(2,1)
e1

(2,1)
q0

1Nice formulae for RPPs
i + j + k −11−q|π |q= ∏∏∏∑
i + j + k −2
1−qi =1 j =1 k =1π ∈RPP((b a ),c)b1cwhere |π | := ∑i,j πi,j.
Gansner’s hook formula [G] (for RPPs with unbounded parts):1|π |∑ q = ∏ 1 − qhλ(i,j)π ∈RPP(λ,∞)(i,j)∈λ(2)11(4,0)e1(4,0)q0

11111111(3)111

11(1,3)q4(2,1)e5(3,1)q4[K2] S. Kamioka, A triple product formula for plane partitions derived frombiorthogonal polynomials, DMTCS proc. BC, 2016 (2016), 671–682.
[M] P. A. MacMahon, Combinatory Analysis, volume 2, Cambridge UniversityPress, Cambridge, 1916.
[G] E. R. Gansner, The Hillman–Grassl correspondence and the enumeration ofreverse plane partitions, J. Combin. Theory Ser. A 30 (1981), 71–89.

1−c−i −k +1+λ0j+k−cq(9a)(9b)

11(1,3)
q5

Remark: The new nice formula (9) refines known nice formulae; the (9)reduces into:
MacMahon’s formula (2) by λ = (b a );Gansner’s hook formula (3) by c → ∞.
See [K3] for further nice formulae derived from Main Theorem.

1(2,1)e6

1(2,2)e61(s,t) (s,t)
The weights qn , en for vertical edges are taken from:

References:[K1] S. Kamioka, Plane partitions with bounded size of parts and biorthogonal
polynomials, arXiv:1508.01674.

∏ ∏1−qc−i −k+1+λ0j+k−c−1

where λi = b and λi0 = a for i < 0, and hλ (i, j) = λi + λ0j − i − j + 1.

(1,3)
e5(2,2)e5(2,1)q5

1(0,4)q4

1

11v` = q

1 − qhλ (i,j−c)∑ v(π ) = ∏ 1 − qhλ(i,j) .
π ∈RPP(λ,c)(i,j)∈λ111,λ0b−` −λ0b−`+1 +1

for reverse place partitions ∈ RPP(λ, c) satisfies a nice formula(0,4)
e4(1,3)e4(2,2)e4(2,1)q4(4,0)e4(4,0)
q3

11

1

1|π |

πi,j(i,j)∈λ k=1

1(0,4)
q3(1,3)q3(2,1)e4(3,1)q3

11u` = qv(π ) = q(0,4)
e3(1,3)e3(2,2)e3(2,1)q3(4,0)e3(4,0)
q2

1

1

1

1(0,4)
q2(1,3)q2(2,1)e3(3,1)q2

1

1

1(0,4)e2(1,3)e2(2,2)e2(2,1)q2(4,0)e2

1,

λ a−` −λ a−`+1 +1Let λ be a Young diagram with a rows and b columns. Let c ≥ 0. Then, theweight

1(0,4)
q1(1,3)q1(2,1)e2(3,1)q1(4,0)q11(2,2)e1(2,1)q1(3,1)q0

MacMahon’s formula [M] (for RPPs of rectangular shape):a(1,3)e1(1,3)q0

hλ ( a,b)

Corollary: New nice formula(0,4)
e1

1

For a Young diagram λ with a rows and b columns letThen, as an instance of Main Theorem:

1(0,4)q0(8)where λi = λ1 = b and λi0 = λ10 = a for i < 0.

Example: For the shape λ = (5, 4, 4, 2, 1) the edge-weight function w islooked as

λ = (λ1, λ2, . . . ): (integer) partition, or a Young diagram.
λ0 = (λ10 , λ20 , . . . ): the Young diagram conjugate with λ.
0hλ (i, j) = λi + λ j − i − j + 1: the legnth of the hook of a cell (i, j) in λ.
RPP(λ, c): the set of RPPs of shape λ with parts ≤ c.

(7b)β[ z ] α : = z α z α +1 · · · z β .

x=qthat is the product of the weights of all the edges passed by P1, . . . , Pc.

(7a)involving parameters x, u1, u2, . . . and v1, v2, . . . wherecv(π ) := ∏ w( Pk ) = ∏ ∏ w(e)(1)

Notations

(s,t)
s+n
st+nq n = [ u ] s +1 (1 − x [ u ] 1 [ v ] 1 ),(s,t)
s + n −1tt+n
e n = x [ u ]1[ v ] 1 (1 − [ v ] t +1 )For the lattice graph we assume an edge-weight function w determined fromthe Young diagram λ, see Example below. We then define the weight v(π ) of
1:1

weakly increasing

(6)

←→P4

A reverse plane partition (RPP) of shape λ is a filling of a Young diagram λwith nonnegative integers which weakly increase along rows and columns.

c∏∏

∑

34 2
4 4 4 24 3 202 1 1 0 0

1:1( a−i,b)qc−kv(π ) =.
( a−i,b−λi )i =1 k =1 q c − kπ ∈RPP(λ,c)
aThe two-dimensional (2D) Toda equation [HTI](s,t+1)(s+1,t)(s,t)(s,t)
qn+ en= q n + e n +1 ,(s,t+1) (s+1,t)(s,t) (s,t)
qne n +1 = q n +1 e n +1(s,t)
2for (s, t) ∈ Z and n = 0, 1, 2, . . . with e0 = 0.

[HTI] R. Hirota, S. Tsujimoto and T. Imai, Difference scheme of soliton
equations, Sūrikaisekikenkyūsho Kōkyūroku, 822, pp. 144–152, 1993.
(5a)
(5b)

[K3] S. Kamioka, Multiplicative partition functions for reverse plane partitions
derived from an integrable dynamical system, Sém. Lothar. Combin. 78B(2017), Article #29, 12 pp.
Acknowledgements: This work was supported by JSPS KAKENHI GrantNo. JP16K05058.