The 1971 Fortuin–Kasteleyn–Ginibre inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008, one of us (Sahi) conjectured an extended version of this inequality for all *n* > 2 monotone functions on a distributive lattice. Here, we prove the conjecture for two special cases: for monotone functions on the unit square in $Rk$ whose upper level sets are *k*-dimensional rectangles and, more significantly, for *arbitrary* monotone functions on the unit square in $R2$. The general case for $Rk,k>2$, remains open.

## I. INTRODUCTION

For functions *f*, *g* on a probability space (*L*, *μ*), their expectation and correlation are defined by

Now suppose further that *L* is a distributive lattice^{8} and that the probability measure *μ* satisfies

In this situation, if *f*, *g* are positive monotone (decreasing) functions^{9} on *L*, then one has

The first inequality is obvious, while the second is the celebrated FKG inequality of Fortuin–Kasteleyn–Ginibre^{1} that plays an important role in several areas of mathematics/physics. We will refer to a distributive lattice *L* with probability measure *μ* satisfying (1.2) as an *FKG poset*.

In formulating (1.2), we have tacitly assumed that the poset *L* is a discrete set. However, the FKG inequality also has important continuous versions, which can be proved by a discrete approximation. For example, if *Q*_{k} = [0, 1]^{k} is the unit hypercube in $Rk$ equipped with the partial order: *x* ≥ *y* if and only if *x*_{i} ≥ *y*_{i} for all *i*, then the FKG inequality holds for the Lebesgue measure and, more generally, for any absolutely continuous measure whose density function satisfies (1.2).

In Ref. 5, Sahi introduced a sequence of multilinear functionals *E*_{n}(*f*_{1}, …, *f*_{n}), *n* = 1, 2, 3, …, generalizing *E*_{1} and *E*_{2} (see Definition 3.1) and proposed the following conjecture:

*(Ref.*

*5*

*, Conjecture 5). If*

*f*

_{1}, …,

*f*

_{n}

*are positive monotone functions on an FKG poset, then*

Sahi^{5} proved the conjecture for the lattice {0, 1} × {0, 1} and for a certain *subclass* of positive monotone functions on the general power set lattice {0, 1}^{k} equipped with a product measure. Since the functionals *E*_{n} satisfy the following “branching” property (Ref. 5, Theorem 6)

the inequalities (1.4) form a *hierarchy* in the following sense: if *C*_{n} denotes the *n*-function positivity conjecture, then *C*_{n} implies *C*_{n−1} for *n* > 2.

The work of Sahi was inspired by that of Richards^{4} who first had the idea of generalizing the FKG inequality to more than two functions. A natural first candidate for such an inequality is the cumulant (Ursell function) *κ*_{n}, but an easy example shows that the inequality already fails for *κ*_{3}. Nevertheless, Richards [Ref. 4, Conjecture 2.5] conjectured the *existence* of such a hierarchy of inequalities, although without an explicit formula for *E*_{n}.

Indeed for *n* = 3, 4, 5, Sahi’s functional *E*_{n} coincides with the “conjugate” cumulant $\kappa n\u2032$ introduced by Richards [Ref. 4, formula (2.2)], although for *n* ≥ 6 one has $En\u2260\kappa n\u2032$. We note also that Ref. 4 contains two “proofs” of the positivity of $\kappa 3\u2032,\kappa 4\u2032,\kappa 5\u2032$—one for a discrete lattice and the other for a continuous analog. However, it seems to us that both proofs have essential gaps. Thus, beyond the special cases treated in Ref. 5, Conjecture 1.1 remains a conjecture, even for *n* = 3, 4, 5.

In this paper, we provide further evidence in support of Conjecture 1.1. We consider the continuous case of the Lebesgue measure on the unit hypercube *Q*_{k} = [0, 1]^{k} in $Rk$, and we prove the inequalities (1.4) for the following two additional cases:

for arbitrary positive monotone functions on the unit square in $R2$ and

for monotone characteristic functions of

*k*-dimensional rectangles in [0, 1]^{k}and, by multlinearity of*E*_{n}, for functions whose level sets are (not necessarily homothetic) rectangles.

First, we treat the case of three functions on $R2$ in Sec. II. This introduces several key ideas, including a reduction to a non-linear inequality involving decreasing sequences. In Sec. III, we define *E*_{n} for arbitrary *n* and prove Conjecture 1.1, first for characteristic functions of *k*-dimensional rectangles and then for general monotone functions on $R2$, that is, we extend Sec. II to all *n* > 3. This requires additional ideas involving the symmetric group *S*_{n} and an intricate induction on *n*. Subsections III A and III B are written in complete generality, and we hope these ideas will help in the eventual resolution of Conjecture 1.1.

Since the FKG inequality has many applications in probability, combinatorics, statistics, and physics, it reasonable to suppose that the generalized inequality will likewise prove to be useful in one or more of these areas. Although we do not have a compelling application in mind, we feel that it is important to find such an application. Indeed, the right application might provide additional insight into Conjecture 1.1 and perhaps even suggest a line of attack.

To end this section, we tantalize the reader with an interesting reformulation of the inequalities *E*_{n} ≥ 0 in terms of a formal power series from Ref. 5. First, if *F*(*x*) is a positive function on a probability space *L*, then it is natural to define the geometric mean of *F* by

Now, suppose *F*(*x*, *t*) is a power series of the form

Then, $logFx,t$ is a well defined power series, and formula (1.6) gives

where the constants *c*_{j} are certain algebraic expressions in various $E(fi1fi2\cdots fip)$.

*(Ref.* *5 **, Conjecture 4). If the* *f*_{1}(*x*), *f*_{2}(*x*), … *is a sequence of positive monotone functions on an FKG poset, then* *c*_{n} ≥ 0 *for all* *n**.*

## II. THE INEQUALITY FOR THREE FUNCTIONS

For three functions, the multilinear functional *E*_{n} introduced in Ref. 5 is given by

We note that *E*_{3} is *different* from the cumulant (Ursell function), which is given by

We will consider the functional *E*_{3} for functions on the unit hypercube,

equipped with the Lebesgue measure and the usual partial order: *x* ≤ *x*′ if and only if $xi\u2264xi\u2032$ for all *i*. We say that a real valued function *f* on *Q*_{k} is **monotone** (decreasing) if *x* ≤ *x*′ implies *f*(*x*) ≥ *f*(*x*′). We note that the FKG inequality is usually stated for monotonically increasing functions, but this is a somewhat arbitrary choice. Indeed, FKG and our theorems for decreasing functions are equivalent to the corresponding results for increasing functions. For a general FKG poset, this follows by reversing the partial order, and for *Q*_{k}, this follows by the change of variables *x*_{i} ↦ 1 − *x*_{i}. We also note that monotonicity for *Q*_{1} has the usual 1-variable meaning of a decreasing function.

*If* *f*, *g*, *h* *are positive monotone functions on* [0, 1]^{2}, *then* *E*_{3}(*f*, *g*, *h*) ≥ 0*.*

The generalization of Theorem 2.1 to *n* functions is given in Theorem 3.6. We now reduce Theorem 2.1 to characteristic functions *χ*_{S}, *S* ⊂ *Q*_{k}. These are defined by *χ*_{S}(*x*) = 1 if *x* ∈ *S* and *χ*_{S}(*x*) = 0 if *x* ∉ *S*. We will say *S* is **monotone** if *χ*_{S} is monotone.

*It suffices to prove Theorem* 2.1 *for* *χ*_{S}, *χ*_{T}, *χ*_{U} *for all monotone* *S*, *T*, *U**.*

Any positive *f* can be written as an integral over the characteristic functions of its upper level sets. Thus, $f(x)=\u222b0\u221e\xi s(x)ds$, with *ξ*_{s}(*x*) = 1 if *f*(*x*) > *s* and 0 otherwise (see the “layer cake principle” in Ref. 2). If *f* is monotone, then *ξ*_{s} is monotone for every *s*. Since *E*_{3} is multi-linear in *f*, *g*, *h*, this reduces Theorem 2.1 to the case of monotone characteristic functions.■

We now describe a further reduction of Theorem 2.1 to a discrete family of characteristic functions. Let $A=A(m)$ be the set of decreasing *m*-tuples of integers, each between 0 and *m*,

For each $a\u2208A$, we define a monotone subset *S*_{a} of *Q*_{2} = [0, 1]^{2} as follows. Divide *Q*_{2} uniformly into *m*^{2} little squares, write *D*_{i,j} for the square with top right vertex (*i*/*m*, *j*/*m*), and set

Then, *S*_{a} is a monotone subset of *Q*_{2}, and, conversely, *any* monotone union of *D*_{i,j} is of this form.

*It suffices to prove Theorem* 2.1 *for* *χ*_{a}, *χ*_{b}, *χ*_{c} *;* $a,b,c\u2208A(m)$; *for all* *m**.*

By Lemma 2.2, it suffices to consider monotone characteristic functions *χ*_{S}, *χ*_{T}, *χ*_{U}. Divide *Q*_{2} uniformly into *m*^{2} little squares *D*_{i,j} as before, and let *S*^{m}, *T*^{m}, *U*^{m} be the unions of the *D*_{i,j} contained in *S*, *T*, *U*, respectively; then, these are monotone subsets of *Q*_{2} of the form (2.5). Moreover, $\chi Sm,\chi Sm\chi Tm$, etc., converge to *χ*_{S}, *χ*_{S}*χ*_{T}, etc., in *L*^{1} as *m* → *∞*. Thus, if $E3(\chi Sm,\chi Tm,\chi Um)\u22650$, then we get $E3(\chi S,\chi T,\chi U)=limm\u2192\u221eE3(\chi Sm,\chi Tm,\chi Um)\u22650$.■

### A. Proof of the three function inequality in two dimensions

We now prove Theorem 2.1 for *χ*_{a}, *χ*_{b}, *χ*_{c}, which suffices by Lemma 2.3. To simplify the notation, we work directly with *a*, *b*, *c* and we define the product *ab*, expectation $E(a)$, etc., as follows:

Then, we have *χ*_{ab} = *χ*_{a}*χ*_{b}, $E(a)=E(\chi a)$, *E*_{2}(*a*, *b*) = *E*_{2}(*χ*_{a}, *χ*_{b}), *E*_{3}(*a*, *b*, *c*) = *E*_{3}(*χ*_{a}, *χ*_{b}, *χ*_{c}).

In particular, by the FKG inequality, we obtain the following lemma:

*For all* *a*, *b* *in* $A$, *we have* *E*_{2}(*a*, *b*) ≥ 0*.*■

To study *E*_{3}(*a*, *b*, *c*), we consider certain perturbations of *a*. We say that $a\u2208A$ has a **descent** at *i* if *a*_{i} > *a*_{i+1}, and in this case, we can define three new sequences *a*^{−} = *a*^{−,i}, *a*^{+} = *a*^{+,i}, *a*^{⋆} = *a*^{⋆,i}, *also* in $A$, in which the following changes, and *only these*, are made to *a*:

*If* *a* *has a descent at* *i**, but* *b* *does not, then we have* $E(a+b)+E(a\u2212b)=2E(ab)$*.*

*b*

_{i}=

*b*

_{i+1}=

*β*, say, then we have

*a*

^{+}

*b*,

*a*

^{−}

*b*, and

*ab*coincide except at

*i*,

*i*+ 1, the result follows.■

*If*

*a*

*has a descent at*

*i*

*, but*

*b*

*and*

*c*

*do not, then*

*If* *a*, *b* *have a descent at* *i* *and* *b*_{i+1} ≤ *a*_{i+1}*, then* *a*^{⋆}*b* = *ab**.*

*j*≠

*i*+ 1, and since

*b*

_{i+1}≤

*a*

_{i+1}, we also have

*a*

^{⋆}

*b*=

*ab*, as claimed.■

*If*

*a*

*and*

*b*

*have a descent at*

*i*

*and*

*b*

_{i+1}≤

*a*

_{i+1}

*, then we have*

*a*

^{⋆}

*b*=

*ab*

*and*

*E*

_{2}(

*b*,

*c*) ≥ 0. Thus, all terms on the right-hand side of (2.17) are positive, which proves the result.■

*For all* *a*, *b*, *c* *in* $A$, *we have* *E*_{3}(*a*, *b*, *c*) ≥ 0*.*

Let $U$ be the set of triples (*a*, *b*, *c*) in $A$ for which *E*_{3}(*a*, *b*, *c*) attains its *minimum*, and let $V$ be the subset of $U$ for which the quantity $E(a)+E(b)+E(c)$ attains its *maximum*.

*a*,

*b*,

*c*are constant sequences. If this is not the case, then

*a*, say, has a descent at some

*i*. If

*b*,

*c*do not have a descent at

*i*, then by Proposition 2.6 we get

*E*

_{3}(

*a*,

*b*,

*c*) ≤

*E*

_{3}(

*a*

^{±},

*b*,

*c*), which forces

*E*

_{3}(

*a*,

*b*,

*c*) =

*E*

_{3}(

*a*

^{±},

*b*,

*c*). Replacing

*a*by

*a*

^{+}, we reach a contradiction since $E(a+)>E(a)$.

If *b*, say, also has a descent at *i*, then by symmetry we may assume *b*_{i+1} ≤ *a*_{i+1}. Then, by Proposition 2.8, *E*_{3}(*a*^{⋆}, *b*, *c*) ≤ *E*_{3}(*a*, *b*, *c*), and we again reach a contradiction since $E(a\u22c6)>E(a)$.

*a*,

*b*,

*c*are constant sequences and, by symmetry, further assume that

*E*

_{3}(

*a*,

*b*,

*c*) = 2

*α*+

*αβγ*− (

*αβ*+

*αβ*+

*αγ*) =

*α*(1 −

*β*) (2 −

*γ*) ≥ 0.■

This proves Theorem 2.1 for *χ*_{a}, *χ*_{b}, *χ*_{c} and, thus, by Lemma 2.3, in general.

## III. THE INEQUALITY FOR *n* FUNCTIONS

### A. The definition of *E*_{n}

In this subsection and Subsection III B, we work with arbitrary functions on a probability space. We start by recalling the definition of the multilinear functional *E*_{n}(*f*_{1}, …, *f*_{n}) from Ref. 5. This involves the decomposition of a permutation *σ* in the symmetric group *S*_{n} as a product of disjoint cycles,

For *σ* as in (3.1), we write *C*_{σ} for the number of cycles in *σ* and we set

Then, the following definition is due to Sahi.^{5}

*f*

^{1}, …,

*f*

^{n}on a probability space

*X*, we define

Using (3.3), one can easily verify that *E*_{1}, *E*_{2}, *E*_{3} coincide with their earlier definitions. We note that the factor of 2 in the term 2*E*(*f*^{1}*f*^{2}*f*^{3}) in formula (2.1) comes from the two 3-cycles (123) and (213). More generally, *E*_{n} will have repeated terms because *E*_{σ} is unchanged if we rearrange the indices within a cycle. For example, for *n* = 4, we have

We now give an explicit formula for *E*_{n} in a special case.

*Let*

*X*= [0, 1]

*be the unit interval equipped with Lebesgue measure, and let*

*f*

^{i}

*be the characteristic function*$\chi [0,ai],0\u2264ai\u22641$

*, with*0 ≤

*a*

_{1}≤ ⋯ ≤

*a*

_{n}≤ 1

*. Then, we have*

We note that the above formula implies that *E*_{n} is positive, i.e., Conjecture 1.1 holds for the Lebesgue measure on [0, 1]. While it is easy enough to give a direct proof the lemma, we prefer to postpone the proof to Subsection III B where we will derive it as a consequence of a more general result.

### B. Algebraic properties of *E*_{n}

We first prove a recursive formula relating *E*_{n} to *E*_{n−1}.

*We have*

*E*

_{n}(

*f*

^{1}, …,

*f*

^{n−1},

*f*) =

*e*

_{1}+ ⋯ +

*e*

_{n−1}−

*e*

_{n},

*where*

*f*=

*f*

^{n}and consider expression (3.3) for

*E*

_{n}(

*f*

^{1}, …,

*f*

^{n}) as a sum over the symmetric group

*S*

_{n}. We decompose

*S*

_{n}as a disjoint union,

*S*

^{(n)}is a subgroup of

*S*

_{n}, naturally isomorphic to

*S*

_{n−1}. By (3.3), we have

^{i}, we consider the map $\sigma \u21a6\sigma \u0304$ defined by dropping

*n*from the cycle decomposition of

*σ*. Thus, for

*n*= 5, we have (13)(245) ↦ (13)(24), (12)(34)(5) ↦ (12)(34), etc. Then, $\sigma \u21a6\sigma \u0304$ defines a bijection from

*each*

*S*

^{(i)}to

*S*

_{n−1}. If

*σ*is in

*S*

^{(i)}and

*i*≠

*n*, then

*i*and

*n*occur in the same cycle of

*σ*, and dropping

*n*does not change the cycle count. This gives

^{i}=

*e*

_{i}. If

*σ*is in

*S*

^{(n)}, then (

*n*) occurs as a separate cycle in

*σ*and we get

^{n}= −

*e*

_{n}. This proves the proposition.■

Lemma 3.2 is now an easy consequence.

*a*

_{i}≤

*a*

_{n}for all

*i*, we get

*f*=

*f*

_{n}, we deduce that

*n*.■

We next establish a useful formula for the partial sum *P*_{c} of *E*_{n} over the set of permutations containing a fixed cycle *c*.

*Let*

*S*

^{c}

*denote the set of permutations*

*σ*∈

*S*

_{n}

*that contain a fixed cycle*

*c*= (

*i*

_{1}, …,

*i*

_{p})

*and let*

*J*

_{c}= {

*j*

_{1},

*j*

_{2}, …} = {1, …,

*n*}\{

*i*

_{1}, …,

*i*

_{p}}

*, then we have*

*S*

^{c}consists of a single permutation if

*p*=

*n*. Otherwise, it consists of permutations of the form

*σ*=

*c*·

*τ*, where

*τ*is a permutation of

*J*

^{c}. Evidently, the number of cycles in

*σ*and

*τ*are related by

*C*

_{τ}=

*C*

_{σ}− 1. Thus, in this case, we have

*τ*.■

### C. Proof of the *n* function inequality for rectangles in any dimension

By a rectangle in dimension *k* or a *k*-rectangle, we mean a subset of [0, 1]^{k} of the form

*If* *f*^{i} *are characteristic functions of* *k**-rectangles, then* *E*_{n}(*f*^{1}, …, *f*^{n}) ≥ 0*.*

*k*≥ 1 and for a given

*k*by induction on

*n*≥ 1. The base cases

*k*= 1 and

*n*= 1 are straightforward and the former by Lemma 3.2. Thus, we may assume

*k*> 1 and

*n*> 1, and we can write

*g*

^{i}is the characteristic function of a (

*k*− 1)-rectangle. By symmetry of

*E*

_{n}, we may assume

*a*

_{i}means that we have

*a*

_{2}= ⋯ =

*a*

_{n}= 1, then we have

*i*> 1 and let

*C*(

*i*) denote all set of all cycles containing

*i*, then we have

*P*

_{c}is as in Proposition 3.4. If

*i*is not minimal in

*c*, then

*P*

_{c}is independent of

*a*

_{i}by (3.10). If

*i*is minimal in

*c*, then 1 ∉

*c*; hence,

*c*has length

*p*<

*n*, and by (3.7) and (3.10), we get

*n*, we have

*b*

_{c}≥ 0 for such

*c*. This means that

*E*

_{n}(

*f*

^{1}, …,

*f*

^{n}) decreases as we increase

*a*

_{2}, …,

*a*

_{n}subject, of course, to condition (3.9). In particular,

*E*

_{n}decreases as we successively increase

*E*

_{n}(

*f*

^{1}, …,

*f*

^{n}) ≥

*a*

_{1}

*E*

_{n}(

*g*

_{1}, …,

*g*

_{n}), which is positive by induction on

*k*.■

If *f* is the characteristic function of a rectangle, then any level set of *f* is either the same rectangle or empty. However, using the layer-cake principle^{2} and multilinearity as in the Proof of Lemma 2.2, we obtain the following immediate extension of the previous result.

*If* *f*^{1}, …, *f*^{n} *are positive, monotone functions whose level sets are (not necessarily homothetic) rectangles, then* *E*_{n}(*f*^{1}, …, *f*^{n}) ≥ 0*.*■

### D. Proof of the *n* function inequality in two dimensions

Our main result is as follows:

*If* *f*^{1}, …, *f*^{n} *are positive and monotone on* [0, 1]^{2}*, then* *E*_{n}(*f*^{1}, …, *f*^{n}) ≥ 0*.*

As before, we can deduce this from the special case of *χ*_{a} as in (2.5).

*It suffices to prove Theorem* 3.6 *for* $\chi a1,\u2026,\chi an$*,* $ai\u2208A(m)$*, for all* *m**.*

This is proved along the same lines as Lemmas 2.2 and 2.3.■

In this section, we work with $A=A(m)$ and to simplify the notation for *a*^{1}, …, *a*^{n} in $A$, we set

Then, we have $En(\chi a1,\u2026,\chi an)=En(a1,\u2026,an)$.

To study the positivity of *E*_{n}, we first consider a special case.

*If*

*a*

^{i}≡

*mα*

_{i}

*are constant sequences with*1 ≤

*α*

_{1}≤ ⋯ ≤

*α*

_{n}≤ 0

*, then*

*L*

_{n}=

*E*

_{n}(

*a*

^{1}, …,

*a*

^{n}). Since

*α*

_{i}≤

*α*

_{n}, we have

*a*

^{i}

*a*

^{n}=

*a*

^{i}for all

*i*. Thus, we get

*n*, the case

*n*= 1 being obvious.■

We now prove the generalization of Proposition 2.6.

*If*

*a*

*has a descent at*

*i*

*, but*

*a*

^{1}, …,

*a*

^{n−1}

*do not, then we have*

We shall prove the next three theorems *together* by induction on *n*.

*If*

*a*

^{1}, …,

*a*

^{n−2},

*b*

*are in*$A$

*;*

*S*

*is a subset of*

*Q*

_{2}

*; and*

*χ*

_{b}

*χ*

_{S}= 0

*, then*

*If*

*a*

^{1}, …,

*a*

^{n−2},

*b*,

*c*

*are in*$A$

*;*

*b*,

*c*

*have a descent at*

*i*

*; and*

*b*

_{i+1}≤

*c*

_{i+1},

*then*

*For all*

*a*

^{1}, …,

*a*

^{n}

*in*$A$,

*we have*

Let us write *A*(*n*), *B*(*n*), and *C*(*n*) for the assertions of Theorems 3.10, 3.11, and 3.12. Then, *A*(1), *B*(1) are vacuously true, while *C*(1) is evident. Therefore, it suffices to prove the implications *A*(*n* − 1) ∧ *C*(*n* − 1) ⟹ *A*(*n*) and *A*(*n*) ⟹ *B*(*n*) ⟹ *C*(*n*) for all *n* ≥ 2.

*A*(

*n*− 1) ∧

*C*(

*n*− 1) ⟹

*A*(

*n*): By assumption, we have

*χ*

_{b}

*χ*

_{S}= 0, and we also have $\chi ai\chi S=\chi Si$, where $Si=S\u2229Sai$. Thus, by Proposition 3.3, we get

*e*

_{n}≥ 0 by C(

*n*− 1) and

*e*

_{n−1}= 0 by (3.12) and (3.13). In addition, $\chi b\chi Si=(\chi b\chi S)\chi ai=0$, and so by symmetry, we can apply

*A*(

*n*− 1) to conclude

*e*

_{i}≤ 0 for

*i*≤

*n*− 2. This implies

*A*(

*n*), (3.16).

*A*(

*n*) ⟹

*B*(

*n*): Define $Sc,Sc\u22c6$ as in (2.5) and put $S=Sc\u22c6\Sc$, then by Lemma 2.7 we have

*A*(

*n*), (3.16), we get $En(\chi a1,\u2026,\chi an\u22122,\chi b,\chi c\u22c6\u2212\chi c)\u22640$, which implies

*B*(

*n*), (3.17).

*B*(*n*) ⟹ *C*(*n*): This argument is similar to the Proof of Theorem 2.9. Let $M$ be the set of *n*-tuples **a** = (*a*^{1}, …, *a*^{n}) in $A$ for which *E*_{n}(**a**) achieves its *minimum*, and let $N$ be the subset of $M$ for which $\lambda (a)=E(a1)+\cdots +E(an)$ achieves its *maximum* on $M$. We claim that for **a** in $N$ each *a*^{i} is a constant sequence; by Proposition 3.8, this clearly implies *C*(*n*), *E*_{n}(**a**) ≥ 0.

If the claim is not true, then one of the sequences has a descent at some *i*. First suppose that only one sequence, by symmetry *a*^{n} = *a*, has a descent at *i*. By Proposition 3.9 and minimality of *E*_{n}(**a**), we deduce *E*_{n}(**a**) = *E*_{n}(*a*^{1}, …, *a*^{n−1}, *a*^{±}). Thus, replacing *a* by *a*^{+} preserves *E*_{n}(**a**) but increases *λ*(**a**), which is a contradiction. If two sequences have a descent at *i*, then by symmetry we may assume these are *a*^{n−1} = *b*, *a*^{n} = *c* with *b*_{i+1} ≤ *c*_{i+1}. Now, *B*(*n*), (3.17), implies that replacing *c* by *c*^{⋆} does not increase *E*_{n}(**a**), but it does increase *λ*(**a**), which is a contradiction.■

This proves Theorem 3.6 for $\chi ai$ and, thus, by Lemma 3.7, in general.

## ACKNOWLEDGMENTS

This work was partially supported by the NSF (Grant Nos. DMS-1939600 and DMS-2001537) and the Simons Foundation (Grant No. 509766). The hospitality of the Institute for Advanced Study is gratefully acknowledged.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX: THE EQUIVALENCE OF CONJECTURES 1.1 AND 1.2

We start by recalling some basic facts about partitions and permutations. For more background and details involving these ideas, we refer the reader to Ref. 3.

A partition *λ* of *n*, of length *l*, is a weakly decreasing sequence of positive integers,

we say that the *λ*_{j} are the *parts* of *λ*, and we write $l\lambda =l$ and $\lambda =n$.

The conjugation action of *S*_{n} permutes the indices in the cycle decomposition (3.1) of an element *σ*. Thus, the class of *σ* is uniquely determined by its “cycle type,” i.e., the partition *λ* whose parts are the cycle lengths of *σ*, arranged in decreasing order. Moreover, if $mi=mi\lambda $ denotes the number of parts of size *i*, then the conjugacy class of cycle type *λ* contains *n*!/*z*_{λ} elements, where

For a function *f* on a probability space, we define its *moments* by the formula

*We have* $En(f,\u2026,f)=n!\u2211|\lambda |=n(\u22121)l(\lambda )\u22121z\lambda \u22121p\lambda (f)$*.*

If *f* is as above and *u* is a parameter, then we can define the formal logarithm

*We have* $expElog(1\u2212uf)=1\u2212\u2211n\u22651unEn(f,\u2026,f)/n!$

*p*

_{k}=

*p*

_{k}(

*f*) and

*p*

_{λ}=

*p*

_{λ}(

*f*) for simplicity, we get

*If*

*f*

_{1},

*f*

_{2}, …

*are functions on a probability space, then we have*

*A*=

*f*

_{1}

*t*+

*f*

_{2}

*t*

^{2}+ ⋯, then by Proposition A.2, we get

*E*

_{n}, we have $En(A,\u2026,A)=\u2211i1,\u2026,inEn(fi1,\u2026,fin)ti1+\cdots in$.■

*For a set of functions* $I$ *on a probability space, the following are equivalent:*

*For all**n**, we have**E*_{n}(*f*_{1}, …,*f*_{n}) ≥ 0*if*$f1,\u2026,fn\u2208I$*.**The power series*$1\u2212expE(log(1\u2212\u2211ifiti))$*has positive coefficients if*$f1,f2,\u2026\u2208I$*.*

*p*

_{1},

*p*

_{2}, …,

*p*

_{n}be the first

*n*primes; define

*s*

_{1}

*k*

_{1}+ ⋯ +

*s*

_{n}

*k*

_{n}=

*N*, where

*s*

_{1}, …,

*s*

_{n}are integers $\u22650$. If some

*s*

_{j}were 0, then

*p*

_{j}would divide the left side but not the right; thus, we must have all

*s*

_{j}> 0 and, hence, that

*s*

_{1}= ⋯ =

*s*

_{n}= 1. Now, it follows from Proposition A.3 that the coefficient of

*t*

^{N}in the power series $1\u2212expE(log(1\u2212\u2211j=1nfjtkj))$ is precisely

*E*

_{n}(

*f*

_{1}, …,

*f*

_{n}). Thus, the second statement implies the first.■

The previous theorem proves the equivalence of Conjectures 1.1 and 1.2. In particular, our Theorem 3.6 implies Conjecture 1.2 for the Lebesgue measure on the unit square in $R2$.

## REFERENCES

A distributive lattice is a partially ordered set, closed under join (supremum) ∨ and meet (infimum) ∧ such that each operation distributes over the other. A key example is the power set of a set, partially ordered by inclusion.

In this paper, we use *positive* as a synonym for *non-negative* and *monotone* for *monotone decreasing*. By reversing the partial order, our results and conjectures hold equally for monotone increasing functions. We note further that the positivity requirement on functions is redundant for the second inequality but essential for the first.