A Note on Covering Young Diagrams with Applications to Local Dimension of Posets

02/21/2019 ∙ by Stefan Felsner, et al. ∙ 0

We prove that in every cover of a Young diagram with 2kk steps with generalized rectangles there is a row or a column in the diagram that is used by at least k+1 rectangles. We show that this is best-possible by partitioning any Young diagram with 2kk-1 steps into actual rectangles, each row and each column used by at most k rectangles. This answers two questions by Kim et al. (2018). Our results can be rephrased in terms of local covering numbers of difference graphs with complete bipartite graphs, which has applications in the recent notion of local dimension of partially ordered sets.

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1 Introduction

Let denote the set of all natural numbers (i.e., positive integers). For we denote to be the set of the first natural numbers. A Young diagram with rows and columns is a subset such that whenever , then provided , as well as provided . A Young diagram111In the literature our Young diagrams are more frequently called Ferrers diagrams. We stick to Young diagram to be consistent with [5]. is visualized as a set of axis-aligned unit squares that are arranged consecutively in rows and columns, each row starting in the first column, and with every row (except the first) being at most as long as the row above. The number of steps of a Young diagram is the number of different row lengths in , i.e., the cardinality of

where elements of are called steps of . Young diagrams with elements, rows, columns, and steps, visualize partitions of the natural number into  unlabeled positive integer summands (summand being the length of row ) with summands on different values and largest summand being .

Figure 1: Left: A Young diagram with rows, columns, and steps. Highlighted are the set of steps (gray), the element (bold boundary), the generalized rectangle (green), and the actual rectangle (orange). Right: The Young diagram with steps and a -local partition of with actual rectangles.

A generalized rectangle in a Young diagram is a set of the form with and and . Note that (unless ) not every set of the form with and satisfies . A generalized rectangle with being a set of consecutive numbers in and being a set of consecutive numbers in is an actual rectangle. A generalized rectangle uses the rows in and the columns in . See the left of Figure 1 for an illustrative example.

Motivated by applications for the local dimension of partially ordered sets, we investigate covering a Young diagram with generalized rectangles such that every row and every column of is used by as few generalized rectangles in the cover as possible. We say that is covered by a set of generalized rectangles if , i.e., is the union of all rectangles in . In this case we also say that is a cover of . If additionally the rectangles in are pairwise disjoint, we call a partition of . For example, the right of Figure 1 shows a Young diagram with a partition into actual rectangles.

Theorem 1.

For any , any Young diagram can be covered by a set of generalized rectangles such that each row and each column of used by at most rectangles in if and only if has strictly less than steps.

We prove Theorem 1 in Section 2, answer the questions raised by Kim et al. in Section 3, and describe the application to local dimension of posets in Section 4.

2 Proof of Theorem 1

Throughout we shall simply use the term rectangle for generalized rectangles, and rely on the term actual rectangle when specifically meaning rectangles that are contiguous. For a Young diagram and , let us define a cover of to be -local if each row of is used by at most rectangles in and each column of is used by at most rectangles in . For , let be the (unique) Young diagram with rows, columns, and steps. See the right of Figure 1.

We start with a lemma stating that instead of considering any Young diagram with steps, we may restrict our attention to just .

Lemma 2.

Let and be any Young diagram with steps. Then  admits an -local cover if and only if admits an -local cover with exactly rectangles.

Proof.

First assume that admits an -local cover . If consists of strictly more than rectangles, then there are , , such that for some step . However, in this case is also an -local cover of with one rectangle less. Thus, by repeating this argument, we may assume that .

Figure 2: Transforming a cover of any Young diagram with steps into a cover of (left) and vice versa (right).

If , there is a row or a column that is not used by any step in . Apply the mapping with

respectively

Intuitively, we cut out row  (respectively column ), moving all rows below one step up (respectively all columns to the right one step left). This gives an -local cover of a smaller Young diagram with steps, and eventually leads to an -local cover of , as desired. See the left of Figure 2.

On the other hand, if admits an -local cover , this defines an -local cover of as follows. Index the rows used by the steps of by and the columns used by the steps of by and let . Defining

for gives an -local cover of . See the right of Figure 2.

Observe that the construction maps an actual rectangle of to an actual rectangle of . Also, if is a partition of , then is a partition of . This will be used in the proof of Item 1 of Theorem 3. ∎

Let us now turn to our main result. In fact, we shall prove the following strengthening of Theorem 1.

Theorem 3.

For any and any Young diagram with steps, the following hold.

  1. [label = ()]

  2. If , then there exists an -local partition of with actual rectangles.

  3. If , then there exists no -local cover of with generalized rectangles.

Proof.

First, let us prove Item 1. For shorthand notation, we define . It will be crucial for us that the numbers solve the recursion

(1)

This follows directly from Pascal’s rule for any with .

Due to Lemma 2 it suffices to show that for any and , there is an -local partition of with actual rectangles.

We define the -local partition by induction on and . For illustrations refer to Figure 3.

Figure 3: Left: The Young diagram with steps and a -local partition of into actual rectangles. Right: The Young diagram with steps, the rectangle with , and the Young diagrams and with and steps, respectively.

If , respectively , then is the set of rows of , respectively the set of columns of . If and , then by (1). Consider the actual rectangle for . Then splits into a right-shifted copy of and a down-shifted copy  of . Note that and .

By induction we have an -local cover of and an -local cover of , each consisting of pairwise disjoint actual rectangles. Define

this is a cover of consisting of pairwise disjoint actual rectangles. Rows to  are used by and at most rectangles in , and rows to are used by at most  rectangles in . Hence each row of is used by at most rectangles in . Similarly each column of is used by at most rectangles in . Thus is an -local partition of by actual rectangles, as desired.

For we obtain an -local partition of by restricting the rectangles of the cover of to the rows from to . This yields an -local partition of a down-shifted copy of .


Now, let us prove Item 2. Due to Lemma 2 it is sufficient to show that for the Young diagram with admits no -local cover. If with has an -local cover, then by restricting the rectangles of the cover to the rows from to we obtain an -local cover of a down-shifted copy of . Therefore, we only have to consider .

Let be a cover of . We shall prove that is not -local. Again, we proceed by induction on and , where illustrations are given in Figure 4.

Figure 4: The Young diagram with steps, the rectangle with , and the Young diagrams and with and steps, respectively.

If , then each row is only used by a single rectangle in , otherwise,  would not be -local. Hence, each row of is a rectangle in . Thus column  of is used by rectangles, proving that is not -local.

The case is symmetric to the previous by exchanging rows and columns.

Now let and . We have . Consider the rectangle for . Then splits into a right-shifted copy of and a down-shifted copy of . Note that .

Let , respectively , be the subset of rectangles in using at least one of the rows in , respectively at least one of the columns in . Note that as each generalized rectangle is contained in .

Prune each rectangle in to the columns and each rectangle in  to the rows . This yields covers of and .

The Young diagram is a copy of and . Hence, by induction the pruned cover is not -local. If some column of is used by at least rectangles in , this column of is used by at least rectangles in , proving that is not -local, as desired. So we may assume that some row  of is used by at least rectangles in .

Symmetrically, is a copy of and . Hence, the pruned is a cover of , which by induction is not -local, and we may assume that some column of is used by at least rectangles in . Hence row in is used by at least rectangles in and column in is used by at least rectangles in . As and element is contained in some rectangle of , either row of is used by at least rectangles or column of is used by at least rectangles (or both), proving that is not -local. ∎

Finally, Theorem 1 follows from Theorem 3 by setting .

3 Local covering numbers

In [5], Kim et al. introduced the concept of covering a Young diagram with generalized rectangles subject to minimizing the maximum number of rectangles in any row or column. Their motivation was to investigate the relations between local difference cover numbers and local complete bipartite cover numbers, which are defined as follows222Deviating from [5], we follow here the terminology and notation of local covering numbers introduced in [6]..

A difference graph is a bipartite graph in which the vertices of one partite set can be ordered in such a way that for , i.e., the neighborhoods of these vertices along this ordering are weakly nesting. Equivalently, a bipartite graph with bipartition , , is a difference graph if admits a bipartite adjacency matrix whose support is a Young diagram :

Then complete bipartite subgraphs of correspond precisely to generalized rectangles in . Rows and columns of correspond to vertices of in and , respectively.

Following the notation in [6], local covering numbers are defined as follows. For a graph class and a graph , an injective -covering of is a set of graphs with . An injective -covering of is -local if every vertex of is contained in at most of the graphs , and the local -covering number of , denoted by , is the smallest for which a -local injective -cover of exists.

Let denote the class of all difference graphs, and the class of all complete bipartite graphs. Clearly, we have for all graphs . Kim et al. [5] asked whether there is a sequence of graphs for which is constant while is unbounded. They prove that for all graphs on vertices,

by showing that whenever is a difference graph with one partite set of size . However, no lower bound on for is established in [5]. Specifically, Kim et al. ask for the exact value of for the difference graph corresponding to the Young diagram . For the case that is a power of they prove the upper bound .

Using Theorem 1 and , we see that

  • for every difference graph the exact value of is the smallest such that for the number of steps333In terms of graphs, this is the number of different sizes of neighborhoods in one partite set. of it holds ,

  • the difference graphs , , defined by Kim et al. satisfy

  • for this sequence of difference graphs is constant , while is unbounded, and

  • for all graphs on vertices,

4 Local dimension of posets

The motivation for Kim et al. [5] to study local difference cover numbers comes from the local dimension of posets, a notion recently introduced by Ueckerdt [9].

For a partially ordered set (short poset) , define a partial linear extension of to be a linear extension of an induced subposet of . A local realizer of is a non-empty set of partial linear extensions such that (1) if in , then in some , and (2) if and are incomparable (denoted ), then in some and in some . The local dimension of , denoted , is then the smallest for which there exists a local realizer of with each appearing in at most partial linear extensions .

For an arbitrary height-two poset , Kim et al. consider the bipartite graph with partite sets and whose edges correspond to the so-called critical pairs:

They prove that

which also gives good bounds for when has larger height, since we have

for the associated height-two poset known as the split of (see [1], Lemma 5.5). Using these results and the ones from the previous section, we can conclude the following for the local dimension of any poset.

Corollary 4.

For any poset on elements with split we have

5 Ferrers Dimension

The aim of this section is to provide some links to research where related things have been investigated with a different terminology.

A Ferrers diagram is a Young diagram. Typically Ferrers diagrams are defined as graphical visualizations of integer partitions.

Riguet [8] defined a Ferrers relation444 According to [4] Ferrers relations have also been studied under the names of biorders, Guttman scales, and bi-quasi-series. as a relation such that

and or .

A relation can be viewed as a digraph with and . A digraph thus corresponding to a Ferrers relation is a Ferrers digraph. Riguet characterized Ferrers digraphs as those in which the sets of out-neighbors are linearly ordered by inclusion. Hence, bipartite Ferrers digraphs are exactly the difference graphs.

By playing with and/or in the definition of a Ferrers relation it can be shown that Ferrers digraphs without loops are 2+2-free and transitive, i.e., they are interval orders. In general, however, Ferrers digraphs are allowed to have loops.

In the spirit of order dimension the Ferrers dimension of a digraph () is the minimum number of Ferrers digraphs whose intersection is . If is poset and the digraph associated with the order relation (reflexivity implies that has loops at all vertices), then . This was shown by Bouchet [2] and Cogis [3], it implies that Ferrers dimension is a generalization of order dimension. Since Ferrers digraphs are characterized by having a staircase shaped adjacency matrix the complement of a Ferrers digraph is again a Ferrers digraph. Therefore, instead of representing a digraph as intersection of Ferrers digraphs containing ( with ). We can as well represent its complement as union of Ferrers digraphs contained in it ( with ). This simple observation is sometimes useful.

The Ferrers dimension of a relation () is the minimum number of Ferrers relations whose intersection is . Note that if is the digraph corresponding to a relation , then . Hence, the result of Bouchet can be expressed as , here we use the notation to emphasize that we interpret the order as a relation. The interval dimension of a poset  is the minimum number of interval orders extending whose intersection is . Interestingly interval dimension is also nicely expressed as a special case of Ferrers dimension: . For this and far reaching generalizations see Mitas [7].

Relations with can be viewed as bipartite graphs. In this setting is the global -covering number of , i.e., minimum number of difference graphs whose union is the bipartite complement of .

We believe that it is worthwhile to study local variants of Ferrers dimension.

Acknowledgments

This research has been mostly conducted during the Graph Drawing Symposium 2018 in Barcelona. Special thanks go to Peter Stumpf for helpful comments and discussions.

References