Interlacing sequences and cylindric partitions

LetO(q, a)be the limit of O_n(q, a)whenngoes to innity. A direct consequence of the Theorem is the following.

Corollary 4.5.5. The generating function for plane overpartitions O(q, a) is

∞

Y

i=1

(1 +aqⁱ)ⁱ⁻¹

(1−qⁱ)^di/2e(1−aqⁱ)^di/2e. We can also get some more general result, as in [BK72]:

Theorem 4.5.6. The generating function of plane overpartitions whose parts lie in a setS of positive integers is:

Y

i∈S







 Q

j∈S(1 +aq^i+j) (1−qⁱ)(1−aqⁱ)

Y

j∈S j<i

1

(1−q^i+j)(1−aq^i+j)









. (4.5.1)

For example

Corollary 4.5.7. The generating function for plane overpartitions into odd parts is

∞

Y

i=1

(1 +aq²ⁱ)ⁱ⁻¹

(1−q²ⁱ⁻¹)(1−aq²ⁱ⁻¹)(1−q²ⁱ)^bi/2c(1−aq²ⁱ)^bi/2c.

For a horizontal strip θ=λ/µwe dene I_λ/µ =

i≥1|θ_i⁰ = 1 andθ_i+1⁰ = 0 J_λ/µ=

n

j≥1|θ_j⁰ = 0 and θ_j+1⁰ = 1 o

. Let

ϕ_λ/µ(t) = Y

i∈I_λ/µ

(1−t^mi(λ)) andψ_λ/µ(t) = Y

j∈J_λ/µ

(1−t^mj(µ)). (4.6.2) Then

ϕ_λ/µ/ψ_λ/µ=bλ(t)/bµ(t).

For an interlacing sequence Λ = (λ¹, . . . , λ^T) with prole A= (A1, . . . , AT−1)we dene Φ_Λ(t):

Φ_Λ(t) =

T−1

Y

i=1

φ_i(t), (4.6.3)

where

φi(t) =







ϕ_λⁱ⁺¹_/λⁱ(t), Ai = 0, ψ_λi/λⁱ⁺¹(t), A_i = 1.

ForΛ = (λ¹, . . . , λ^T) and M = (µ¹, . . . , µ^S) such thatλ^T =µ¹ we dene

Λ·M = (λ¹, . . . , λ^T, µ², . . . , µ^S) and

Λ∩M =λ^T. Then

AΛ·M = AΛAM

bΛ∩M and ΦΛ·M = ΦΛΦM. (4.6.4)

For an interlacing sequence Λ = (λ¹, . . . , λ^T) with prole A= (A1, . . . , AT−1)we dene the reverseΛ = (λ^T, . . . , λ¹) whose prole isA= (1−A_T−1, . . . ,1−A₁). Then

A_Λ=A_Λ and Φ_Λ= b_λ1Φ_Λ

b_λ^T . (4.6.5)

For an ordinary partition λ we construct an interlacing sequence hλi = (∅, λ¹, . . . , λ^L) of length L+ 1 =l(λ) + 1, where λⁱ is obtained fromλby truncating the last L−iparts.

Then

A_hλi= Φ_hλi=b_λ. (4.6.6)

In our earlier paper [Vul09] (Propositions 2.4 and 2.6) we have shown that for a plane partition Π

Φ_Π=A_Π. (4.6.7)

Now, more generally

Proposition 4.6.1. If Λ = (λ¹, . . . , λ^T) is an interlacing sequence then

Φ_Λ= A_Λ b_λ¹.

Proof. If we show that the statement is true for sequences with constant proles then induc- tively using (4.6.4) we can show it is true for sequences with arbitrary prole. It is enough to show that the statement is true for sequences with (0, . . . ,0) prole because of (4.6.5).

So, let Λ = (λ¹, . . . , λ^T) be a sequence with(0, . . . ,0) prole. Then Π = λ¹

·Λ· hλ^Ti is a plane partition and from (4.6.4), (4.6.5), (4.6.6) and (4.6.7) we obtain thatAΠ=AΛ and Φ_Π=b_λ¹Φ_Λ.

For skew plane partitions and cylindric partitions we obtain the following two corollaries.

Corollary 4.6.2. For a skew plane partition Π we have thatΦΠ=AΠ.

Corollary 4.6.3. If Π is a cylindric partition given with Λ = (λ⁰, . . . , λ^T) then ΦΛ=A^cyl_Π . The last corollary comes from the observation that if a cylindric partition Πis given by a sequenceΛ = (λ⁰, . . . , λ^T) then

A^cyl_Π (t) =A_Λ(t)/b_λ0(t). (4.6.8) In the rest of this Section we prove generalized MacMahon's formulas for skew plane partitions and cylindric partitions that are stated in Theorems 4.1.7 and 4.1.8. The proofs of these theorems were inspired by [Bor07], [OR03] and [Vul09]. We use a special class of symmetric functions called Hall-Littlewood functions.

4.6.1 The weight functions

In this Subsection we introduce weights on sequences of ordinary partitions. For that we use Hall-Littlewood symmetric functionsP and Q. We recall some of the facts about these functions, but for more details see Chapters III and VI of [Mac95]. We follow the notation used there.

Recall that Hall-Littlewood symmetric functions P_λ/µ(x;t) and Q_λ/µ(x;t) depend on a parametertand are indexed by pairs of ordinary partitionsλandµ. In the case whent= 0 they are equal to ordinary Schur functions and in the case when t=−1 to SchurP and Q functions.

The relationship between P and Q functions is given by (see (5.4) of [Mac95, Chapter III])

Q_λ/µ = bλ

bµ

P_λ/µ, (4.6.9)

wherebis given with (4.6.1). Recall that (by (5.3) of [Mac95, Chapter III] and (4.6.9))

P_λ/µ =Q_λ/µ = 0 unlessλ⊃µ. (4.6.10) We set P_λ =P_λ/∅ and Q_λ=Q_λ/∅. Recall that ((4.4) of [Mac95, Chapter III])

H(x, y;t) :=X

λ

Q_λ(x;t)P_λ(y;t) =Y

i,j

1−tx_iy_j 1−x_iy_j .

A specialization of an algebra A is an algebra homomorphism ρ : A →C. If ρ and σ are specializations of the algebra of symmetric functions then we writeP_λ/µ(ρ;t),Q_λ/µ(ρ;t) and H(ρ, σ;t) for the images of P_λ/µ(x;t), Q_λ/µ(x;t) and H(x, y;t) under ρ, respectively ρ⊗σ. Every map ρ : (x1, x2, . . .) → (a1, a2, . . .) where ai ∈ C and only nitely many a_i's are nonzero denes a specialization. These specializations are called evaluations. A multiplication of a specializationρby a scalara∈Cis dened by its images on power sums:

pn(a·ρ) =aⁿpn(ρ).

If ρ is a specialization ofΛ wherex1 =a, x2 =x3 =. . .= 0 then by (5.14) and (5.14')

of [Mac95, Chapter VI]

Q_λ/µ(ρ;t) =







ϕ_λ/µ(t)a^|λ|−|µ| λ⊃µ,λ/µis a horizontal strip,

0 otherwise.

(4.6.11)

Similarly,

P_λ/µ(ρ;t) =







ψ_λ/µ(t)a^|λ|−|µ| λ⊃µ,λ/µis a horizontal strip,

0 otherwise.

(4.6.12)

Let T ≥2 be an integer and ρ^±= (ρ^±₁, . . . , ρ^±_T−1) be nite sequences of specializations.

For a sequence of partitions Λ = (λ¹, . . . , λ^T) we set the weight functionW(Λ;t) to be

W(Λ) = q^T|λT|X

M T−1

Y

n=1

P_λⁿ_/µⁿ(ρ⁻_n;t)Q_λn+1/µⁿ(ρ⁺_n;t),

where q and t are parameters and the sum ranges over all sequences of partitions M = (µ¹, . . . , µ^T−1).

We can also dene the weights using another set of specializationsR^±= (R^±₁, . . . , R_T^±₋₁) were R^±_i =q^±iρ^±_i . Then

W(Λ) =X

M

W(Λ, M, R⁻, R⁺), where

W(Λ, M, R⁻, R⁺) =q^|Λ|

T−1

Y

n=1

P_λⁿ_/µⁿ(R⁻_n;t)Q_λⁿ⁺¹_/µⁿ(R⁺n;t).

We will focus on two special sums

Z_skew = X

Λ=(∅,λ¹,...,λ^T,∅)

W(Λ)

and

Z_cyl= X

Λ=(λ⁰,λ¹,...,λ^T) λ⁰=λ^T

W(Λ).

4.6.2 Specializations

We show that for a suitably chosen specialization the weight function vanishes for every sequence of ordinary partitions unless this sequence represents an interlacing sequence, in which case it becomes (4.6.3).

Let A⁻ = (A⁻₁, . . . , A⁻_T−1) and A⁺ = (A⁺₁, . . . , A⁺_T−1) be sequences of 0's and 1's such thatA⁻_k +A⁺_k = 1.

If specializations R^± are evaluations given by

R^±_k :x₁=A^±_k, x₂ =x₃ =. . .= 0 (4.6.13) thenW(Λ)vanishes unlessΛ is an interlacing sequence of prole A⁻ and in that case

W(Λ;t) = ΦΛ(t)q^|Λ|. Then from Corollaries 4.6.2 and 4.6.3 we have that

Z_skew= X

Π∈Skew(T,A)

A_Π(t)q^|Π|

and

Z_cyl= X

Π∈Cyl(T,A)

A^cyl_Π (t)q^|Π|

whereA⁻ in both formulas is given by the xed proleA of skew plane partitions, respec- tively cylindric partitions.

4.6.3 Partition functions

Ifρ⁺ isx1=s, x2 =x3=. . .= 0 and ρ⁻ is x1 =r, x2 =x3 =. . .= 0 then

H(ρ⁺, ρ⁻) = 1 +tsr 1−sr. Thus, for specializations given by (4.6.13)

H(ρ⁺_i , ρ⁻_j ) = 1 +tq^j−iA⁺_i A⁻_j

1−q^j−iA⁺_i A⁻_j . (4.6.14)

We have shown in our earlier paper (Proposition 2.2 of [Vul09]) that Proposition 4.6.4.

Z_skew(ρ⁻, ρ⁺) = Y

0≤i<j≤T

H(ρ⁺_i , ρ⁻_j ).

Then this proposition together with (4.6.14) implies Theorem 4.1.7. The generating formula for skew plane partitions can also be seen as the generating formula for reverse plane partitions as explained in Introduction.

Each skew plane partition can be represented as an innite sequence of ordinary parti- tions by adding innitely many empty partitions to the left and right side. That way the proles become innite sequences of 0's and 1's. Theorem 4.1.7 also gives the generating formula for skew plane partitions of innite prolesA= (. . . , A−1, A₀, A₁, . . .):

X

Π∈Skew(A)

A_Π(t)q^|Π|= Y

i<j Ai=0, Aj=1

1−tq^j−i

1−q^j−i. (4.6.15)

Similarly for cylindric partitions, using (4.6.14) together with the following proposition we obtain Theorem 4.1.8.

Proposition 4.6.5.

Z_cyl(R⁻, R⁺) =

∞

Y

n=1

1 1−q^nT

T

Y

k, l=1

H(q^{(k−l)(T)+(n−1)T}R⁻_k, R⁺_l ), (4.6.16)

where i_(T) is the smallest positive integer such thati≡i_(T) mod T. Proof. We use

X

λ

Q_λ/µ(x)P_λ/ν(y) =H(x, y)X

τ

Q_ν/τ(x)P_µ/τ(y). (4.6.17) The proof of this is analogous to the proof of Proposition 5.1 that appeared in our earlier paper [Vul07]. Also, see Example 26 of Chapter I, Section 5 of [Mac95].

The proof of (4.6.16) uses the same idea as in the proof of Proposition 1.1 of [Bor07].

We start with the denition ofZ_cyl(R⁻, R⁺). We omit index cyl.

Z(R⁻, R⁺) = X

Λ,M

W(Λ, M, R⁻, R⁺) =X

Λ,M

q^|Λ|

T

Y

n=1

P_λⁿ⁻¹_/µⁿ(R⁻_n)Q_λⁿ_/µⁿ(R_n⁺)

= X

Λ,M

q^|µ|

T

Y

n=1

P_λⁿ⁻¹_/µn(qR⁻_n)Q_λⁿ_/µⁿ(R⁺_n).

If x = (x1, x2, . . . , xT) is a vector we dene the shift as sh(x) = (x2, . . . , xT, x1). We set R₀^±=R_T^±,µ₀=µ_T andτ₀ =τ_T. If using formula (4.6.17) we substitute sums overλⁱ's with sums over τⁱ⁻¹'s we obtain

Z(R⁻, R⁺) = H(qshR⁻;R⁺)X

µ,τ

q^|µ|Q_µ¹_/τ⁰(R_T⁺), P_µ⁰_/τ⁰(qR⁻₁)·

·Q_µ2/τ¹(R⁺₁)P_µ¹_/τ¹(qR⁻₂)· · ·Q_µ⁰_/τ^T−1(qR⁺_T−1)P_µ^T−1_/τ^T−1(R⁻_T)

= H(qshR⁻;R⁺)X

µ,τ

W(shµ, τ, qshR⁻, R⁺)

= H(qshR⁻;R⁺)Z(qshR⁻, R⁺).

Sincesh^T = id if we apply the same trickT times we obtain

Z(R⁻, R⁺) =

T

Y

i=1

H(qⁱshⁱR⁻;R⁺)·Z(q^TR⁻, R⁺)

=

T

Y

i=1

H(qⁱshⁱR⁻;R⁺)·Z(sR⁻, R⁺).

wheres=q^T. Thus,

Z =

∞

Y

n=1 T

Y

i=1

H(q^i+(n−1)T shⁱR⁻;R⁺) lim

n→∞Z(sⁿR⁻, R⁺).

Because

n→∞lim Z(sⁿR⁻, R⁺) = lim

n→∞Z(trivial, R⁺) =

∞

Y

n=1

1 1−sⁿ

and

T

Y

i=1

H(qⁱshⁱR⁻, R⁺) =

T

Y

l=1

" _T Y

k=l+1

H(q^k−lR⁻_k, R⁺_l )

l

Y

k=1

H(q^T+k−lR⁻_k, R⁺_l )

#

=

T

Y

k, l=1

H(q^(k−l)(T)R⁻_k, R⁺_l ),

we conclude that (4.6.16) holds.

Observe that if in Theorem 4.1.8 we let T → ∞, i.e., the circumference of cylinder goes to innity then we reconstruct (4.6.15).