When will the Stanley depth increase

Let $I\subset S=\KK[x_1,...,x_n]$ be an ideal generated by squarefree monomials of degree $\ge d$. If the number of degree $d$ minimal generating monomials $\mu_d(I)\le \min(\binom{n}{d+1},\sum_{j=1}^{n-d}\binom{2j-1}{j})$, then the Stanley depth $\sdepth_S(I)\ge d+1$.


Introduction
Throughout this paper, let K be a field and S = K[x 1 , . . . , x n ] a polynomial ring in n variables over K. The ring S has a natural Z n -grading. If M is a finitely generated Z n -graded S-module, a Stanley decomposition of M is a finite direct sum decomposition If we consider isomorphism instead of equality in the previous Stanley decomposition P, we will land up in the notion of Hilbert depth, which is the main topic of [BKU10].
The driving force for investigating the Stanley depth of a finitely generated Z ngraded module M is the conjecture raised by Stanley [Sta82], which says This conjecture will imply ([HJY08, 4.5]) that Cohen-Macaulay simplicial complexes are partitionable, which was separately conjectured by Garsia [Gar80,5.2] and Stanley [Sta79,p. 149].
To have an insight into the properties of Stanley depth, one lacks the many powerful tools as those for the normal algebraic depth. Deciding the Stanley depth of interesting modules is already a headache for researchers. Currently, the Stanley depth is known only for a very narrow scope of modules, the overwhelming majority of which has equality in the Stanley conjecture ( †).
The paper [HVZ09] by Herzog, Vladoiu and Zheng was a breakthrough along this line. Their method attacks the problem of computing the Stanley depth sdepth(I/J) for monomial ideals J ⊂ I in S. This method, though not a panacea, contributes fundamentally to the knowledge of Stanley decompositions from both theoretical and computational perspectives. For instance, based on this method, Biró et al. [BHK + 10] can show that sdepth S ( x 1 , . . . , x n ) = n 2 . Notice that depth S ( x 1 , . . . , x n ) = 1. Other nontrivial computations and estimates can be found in, for instance, [KY09], [Oka11], [She09] and their references.
Throughout this paper, I will be a monomial ideal in S, generated by squarefree monomials of degree ≥ d. The task of the current paper is to investigate when will sdepth(I) ≥ d + 1. Our main result is the following theorem.
Theorem 1.1. Let I ⊂ S = K[x 1 , . . . , x n ] be an ideal generated by squarefree monomials of degree ≥ d. If the number of degree d minimal generating monomials then the Stanley depth sdepth S (I) ≥ d + 1.
Let us finish this introduction by going over the structure of this paper. In section 2, we will go over Herzog, Vladoiu and Zheng's method for computing the Stanley depth of monomial ideals. We will tailor it to the squarefree case and prove a special case of the main theorem. In section 3, we will inspect several combinatorial constructions, which are essential for deciding when will the Stanley depth increase. In the final section, we will complete the proof and provide additional remarks and questions.

Herzog, Vladoiu and Zheng's method
By convention, we denote the set { 1, 2, . . . , n } by [n]. For the squarefree monomial ideal I, consider the associated set This is a partially ordered set (poset) with respect to inclusion. When A, B ∈ P I , the interval [A, B] is the set { C ∈ P I : A ⊂ C ⊂ B }. Herzog, Vladoiu and Zheng's method [HVZ09, 2.5] for squarefree monomial ideals can be easily checked to be equivalent to the following characterization: Lemma 2.1. Let k be a positive integer. Then sdepth(I) ≥ k if and only if P I has a disjoint partition P : This can be further simplified. Consider the reduced associated poset Remark 2.3. Let I ⊂ J be two S-ideals which are generated by squarefree monomials of degree d. If sdepth(J) ≥ d + 1, then P d+1 J is partitionable by ( ‡). The restriction of such a partition to P d+1 I shows that P d+1 I is also partitionable. Thus, sdepth(I) ≥ d + 1. Notice that in general, we cannot compare sdepth(I) with sdepth(J) even if there exists containment between I and J. For instance, for the three squarefree monomial ideals Corollary 2.4. Suppose I is generated by squarefree monomials of degree ≥ d and sdepth(I) ≥ d + 1, then the number of degree d minimal generators µ d (I) ≤ n d+1 .
Proof. Since sdepth(I) ≥ d + 1, P d+1 I is partitionable and has a partition P d+1 When n ≥ 2d + 1, we have µ d (I) ≤ n d ≤ n d+1 . Thus Corollary 2.4 does not provide much information in this case. However, we have Proposition 2.5. Suppose n ≥ 2d + 1 and I is generated by squarefree monomials of degree ≥ d. Then sdepth(I) ≥ d + 1.
Proof. Recall that the squarefree Veronese ideal I n,d is the ideal generated by all degree d squarefree monomials of S = K[x 1 , . . . , x n ]. It follows from [KSSY11, 1.1] that sdepth(I n,d ) ≥ d + 1. Now, we use Remarks 2.2 and 2.3.
For each squarefree monomial m = x i1 · · · x i k ∈ S, we denote the set { 1, . . . , n }\ { i 1 , . . . , i k } by m ∁ . Suppose I is a squarefree monomial S-ideal and G(I) is the set of minimal generating monomials of I. We call the simplicial complex ∆ ∁ (I) := m ∁ : m ∈ G(I) the complement complex of I. For each simplicial complex over [n], there is a unique squarefree monomial ideal I such that ∆ = ∆ ∁ (I). Thus, we will call I the complement ideal of ∆. It is clear that I is generated by its degree k part I k if and only if ∆ ∁ (I) is pure of dimension n − k − 1. When ∆ ∁ (I) is pure, the number of facets f n−k−1 (∆ ∁ (I)) = µ(I).
Now, let I be a squarefree monomial ideal which is pure of degree d. We will relate the reduced associated poset P d+1 F , such that all these F 's are pairwise distinct. The third condition is closely related to the problem of finding systems of distinct representatives (SDR). It provides the framework for our further investigation. In the following, we will call a pure simplicial complex ∆ uniformly collapsible if it satisfies the third condition above. It is straightforward to see that if ∆ is a uniformly collapsible complex of dimension δ − 1, then f δ−2 ≥ f δ−1 . Here, f (∆) = (f −1 = 1, f 0 , . . . , f δ−1 ) is the f -vector of ∆. Actually, we have the following characterization: Lemma 3.3. For any (δ − 1)-dimensional pure simplicial complex ∆, the following two conditions are equivalent: (a) The complex ∆ is uniformly collapsible; Proof. For the pure complex ∆, we consider its associated bipartite graph G defined as follows. The vertex set is V (G) = X ∪ Y where X is the set of all (δ − 1)dimensional faces (facets) of ∆, while Y is the set of all (δ − 2)-dimensional faces of ∆. An edge of G has endpoints x ∈ X and y ∈ Y if and only if x ⊃ y in ∆. We will use Γ(x) to denote the set of all vertices adjacent to a given vertex x ∈ X.
If A is a subset of X, we denote by Γ  Before we proceed to the next technical lemma, we need reviewing one nice combinatorial interpretation of the Catalan numbers C n := 1 n+1 2n n = 2n n − 2n n+1 for n ≥ 0.
Remark 3.5 ([vLW01, 14.8]). Consider walks in the X-Y plane where each step is U : (x, y) → (x + 1, y + 1) or D : (x, y) → (x + 1, y − 1). Let A = (0, k) and B = (n, m) be two integral points on the upper halfplane. It follows from the André's reflection principle that there are n l2 − n l1 paths from A to B that do not meet the X-axis. Here, 2l 1 = n − k − m and 2l 2 = n − m + k. As a result, there are C n−1 paths from (0, 0) to (2n, 0) in the upper halfplane that do not meet the X-axis between these two points. Furthermore, if we allow the paths to meet the X-axis without crossing, then the number is C n .
With respect to the Macaulay representation (1), we define Lemma 3.6. For any positive integer x such that x ≤ ξ k := k j=1 Proof. Suppose (1) gives the Macaulay representation of x. We need to show k j=i a j j − 1 ≥ k j=i a j j .
In view of Lemma 3.1, we obtain a k ≤ 2k − 1. If a k = 2k − 1, we can consider the case where k ′ = k − 1 and 2j−1 j . The conclusion will follow from the induction on k, with the case k = 1 being trivial.
Thus we may assume that a k < 2k − 1. Let k 0 be the smallest integer such that for all k 0 ≤ j ≤ k we have a j < 2j − 1. Now, it suffices to prove First of all, let us look at the summand on the left hand side of the inequality (2). By our choice of k 0 , we have k 0 > 1 and a k0−1 ≥ 2k 0 − 3. Thus, for j = k 0 , k 0 + 1, . . . , k, we have a j ≥ j + k 0 − 2. When a j < 2j − 1, the integer is the number of paths in the X-Y plane from A = (0, 1) to B j,aj = (a j , 2j − 1 − a j ) that do not meet the X-axis. In particular, this is a positive integer. When a j < 2j − 2, any such a path followed by a step D as in Remark 3.5 gives a path from A to B j,aj +1 . Thus, (3) is an increasing function for a j ∈ { j + k 0 − 2, j + k 0 − 1, . . . , 2j − 2 }. Now the infimum of the left hand side of (2) is achieved when a j = j + k 0 − 2. Henceforth, without loss of generality, we may assume that k = k 0 and a k = 2k − 2, whence a k−1 = 2k − 3.
Next, let us consider the summand on the right hand side of the inequality (2). Notice that a k = 2k − 2, thus a j ≤ k − 2 + j. Now we have which is positive only when a j ≥ 2j − 1. When this condition is indeed satisfied, the integer (4) is the number of paths in the X-Y plane from A = (0, 1) to B j,aj = (a j , a j + 1 − 2j) that do not meet the X-axis. Any such a path followed by a step U as in Remark 3.5 gives a path from A to B j,aj +1 . Thus, (4) is an increasing function for a j ∈ { 2j − 1, 2j, . . . , k − 2 + j }. Now the supremum of the right hand side of (2) is achieved when i = 1 and a j = k − 2 + j for j = 1, . . . , k − 1. Now it suffices to prove As a matter of fact, we have One can also explain this difference being 1 by the paths argument in Remark 3.5.
Next, consider the following property ( * ): If ∆ is a pure simplicial complex of dimension δ − 1 and f δ−1 (∆) ≤ f δ−2 (∆), then ∆ is uniformly collapsible. For investigating this property, we have to be equipped with further apparatus. We will need the following fact from [Duv94,p79]. Define the reverse lexicographical order ≤ rlex on the k-subsets of [n] := { 1, 2, . . . , n } as follows. Let S = { i 1 < · · · < i k } and T = { j 1 < · · · < j k } be two k-subsets. We say S < rlex T if for some q, we have i q < j q and i p = j p for p > q. A collection C of k-subsets of [n] is compressed if S < rlex T and T ∈ C imply S ∈ C. Since ≤ rlex is a total ordering, there is only one compressed collection of k-subsets of size l, 1 ≤ l ≤ n k . We will call it C l n,k and denote the (k − 1)-dimensional simplicial complex C l n,k by ∆ n,k l . The complement ideal of ∆ n,k l will be written as I l n,n−k . It is generated by l squarefree monomials of degree n − k. For 1 ≤ d ≤ n and l = n d , the ideal I l n,d is the usual squarefree Veronese ideal I n,d . The shadow of any collection C of k-subsets is The shadow ∂C l n,k is also compressed and ∂C l n,k = ∂ k−1 (l). The proof of this fact can be found, for instance, in [GK78,Section 8]. This implies that f k−2 (∆ n,k l ) = ∂ k−1 (f k−1 (∆ n,k l )) = ∂ k−1 (l). When ∆ is pure of dimension δ − 1 and C is the set of all facets, then ∂C is the set of all (δ − 2) faces. In general, we will have f δ−2 (∆) ≥ ∂ δ−1 (f δ−1 (∆)), namely |∂C| ≥ ∂ δ−1 (|C|); see [GK78,8.1].
However, the property ( * ) does not hold in general.
In the current context, we always assume that n/2 ≤ d ≤ n, whence 2δ ≤ n. The obstacle in the previous example is created by introducing extra vertices; now the number of vertices is at least 3δ − 1. Thus, we are interested in the following question: Question 3.10. Fix the degree difference δ. If n = 2δ, does the property ( * ) hold? If the answer is positive, what is the largest integer n < 3δ − 1 such that ( * ) holds?

Proof of Theorem 1.1
We have gathered all the apparatus for proving the main theorem.
Proof. By virtue of Remark 2.2, we may assume that I is pure of degree d. For 1 ≤ d < n, write δ = n − d for the difference of degrees.
Remark 4.1. We want to emphasize that the condition in Theorem 1.1 is optimal. With δ = n − d, there is not much to mention for the case n ≤ 2δ − 1. When n ≥ 2δ, we will take I = I ξ δ +1 n,d . It has been manifested in Example 3.7 that the complement complex ∆ n,δ ξ δ +1 is not uniformly collapsible, whence sdepth S (I) = d. We finish by noticing that µ d (I) = ξ δ + 1.
Remark 4.2. When n/2 ≤ d < n, the set Ξ := { I ⊂ S | I is pure of degree d and sdepth(I) = d } is non-empty and partially ordered with respect to inclusion. If I ∈ Ξ is minimal, then µ(I) ≥ ξ δ + 1. This inequality can be strict if the dimension n is not too small relative to the difference δ = n − d. We will only show this in the special case when d = n − 2. Let G be the graph on [n] (1-dimensional pure simplicial complex) with edges It is a circle with a chord. All 1-dimensional proper subcomplexes of G are uniformly collapsible, while G itself is not. Let I be the degree n − 2 complement ideal of the complex G. It satisfies that sdepth(I) = n − 2 and µ(I) = n + 1. Furthermore, this ideal is minimal in Ξ.
Since n + 1 is smaller when compared with n d or n d+1 in this situation, we are interested in  is also nonempty. For any I ∈ Ξ ∁ , we have µ(I) ≤ n d+1 . We will show that max µ(I) I ∈ Ξ ∁ = n d + 1 .
Suppose k is an integer with 1 ≤ k ≤ n − 1. If a squarefree monomial ideal I is pure of degree k and sdepth(I) ≥ k + 1, we have a union of disjoint intervals Here, x m stands for x i1 x i2 · · · x i k if m = { i 1 , . . . , i k }. Now, simply set J = x m ∁ | m ∈ G(I) . The squarefree monomial ideal J is pure of degree n − k − 1 and sdepth(J) ≥ n − k. This correspondence from I to J, though not one-to-one, preserves the minimal number of generators. Now, we are reduced to show the existence of a squarefree monomial ideal J that is pure of degree n−d−1 with µ(J) = n d+1 and sdepth(J) ≥ n−d. This monomial ideal J has to be the squarefree Veronese ideal I n,n−d−1 . Since 2d ≥ n − 1, it has the desired properties.
Notice that any set of squarefree monomials has a squarefree shadow; see [BEOS09,2.2]. Thus, we can prove Theorem 1.1 directly, without resorting to the complement complex. However, we find this approach less intuitive, especially during the construction of the simplicial complex ∆ in Example 3.9 and the graph G in Remark 4.2.