Arc Permutations

Arc Permutations

Abstract

Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern avoidance. It is also shown that arc permutations carry a natural affine Weyl group action, and that the number of geodesics between a distinguished pair of antipodes in the associated Schreier graph, as well as the number of maximal chains in the weak order on unimodal permutations, are both equal to twice the number of standard Young tableaux of shifted staircase shape. Finally, a bijection from non-unimodal arc permutations to Young tableaux of certain shapes, which preserves the descent set, is described and applied to deduce a conjectured character formula of Regev.

1 Introduction

A permutation in the symmetric group is an arc permutation if every prefix forms an interval in . It was found recently that arc permutations play an important role in the study of graphs of triangulations of a polygon [3]. A familiar subset of arc permutations is that of unimodal arc permutations, which are the permutations whose inverses have one local maximum or one local minimum. These permutations appear in the study of Hecke algebra characters [4, 14]. Their cycle structure was studied by Thibon [17] and others.

In this paper we study combinatorial properties and structures on these sets of permutations.

In Section 3 it is shown that both arc and unimodal permutations may be characterized by pattern avoidance, as described in Theorem 3.2 and Proposition 3.4.

In Section 4 we describe a bijection between unimodal permutations and certain shifted shapes. The shifted shape corresponding to a unimodal permutation has the property that standard Young tableaux of that shape encode all reduced words of . It follows that

  • Domination in the weak order on unimodal permutations is characterized by inclusion of the corresponding shapes (Theorem 5.1). Hence, this partially ordered set is a modular lattice (Proposition 5.3).

  • The number of maximal chains in this order is equal to twice the number of staircase shifted Young tableaux, that is, (Corollary 5.5).

The above formula is analogous to a well-known result of Richard Stanley [16], stating that the number of maximal chains in the weak order on is equal to the number of standard Young tableaux of triangular shape.

In Section 6 we study a graph on arc permutations, where adjacency is defined by multiplication by a simple reflection. It is shown that this graph has the following property: an arc permutation is unimodal if and only if it appears in a geodesic between two distinguished antipodes. Hence the number of geodesics between these antipodes is, again, . This result is analogous to [3, Theorem 9.9], and related to [10, Theorem 2].

The set of non-unimodal arc permutations is not a union of Knuth classes. However, it carries surprising Knuth-like properties, which are described in Section 7. A bijection between non-unimodal arc permutations and standard Young tableaux of hook shapes plus one box is presented, and shown to preserve the descent set. This implies that for ,

where denotes the set of non-unimodal arc permutations in , denotes the set of standard Young tableaux of shape for some , and is the descent set of (see Theorem 7.7). Further enumerative results on arc permutations by descent sets appear in Section 8. These enumerative results are then applied to prove a conjectured character formula of Amitai Regev in Section 9.

Interactions with other mathematical objects are discussed in the last two sections: close relations to shuffle permutations are pointed out in Section 11; further representation theoretic aspects are discussed in Section 10. In particular, Section 10.1 studies a transitive affine Weyl group action on the set of arc permutations, whose resulting Schreier graph is the graph studied in Section 6.

2 Basic concepts

In the following definitions, an interval in is a subset for some , and an interval in is a subset of the form or for some .

2.1 Unimodal permutations

Definition 2.1.

A permutation is left-unimodal if, for every , the first letters in form an interval in . Denote by the set of left-unimodal permutations in .

  • The permutation is left-unimodal, but is not.

    Claim 2.2.

    .

    Proof.

    A left-unimodal permutation is uniquely determined by the subset of values such that . There are such subsets. ∎

    We denote by the descent set of a permutation , and by the pair of standard Young tableaux associated to by the RSK correspondence. For a standard Young tableau , its descent set is defined as the set of entries that lie strictly above the row where lies. It is well known that if , then and .

    Remark 2.3.

    A permutation is left-unimodal if and only if for some . In other words if and only if , where is a hook with entries in the first column, and is any hook with the same shape as . It follows that left-unimodal permutations are a union of Knuth classes.

    Definition 2.4.

    A permutation is unimodal if one of the following holds:

    • every prefix forms an interval in ; or

    • every suffix forms an interval in .

    Denote by the set of unimodal permutations in .

    We remark that our definition of unimodal permutations is slightly different from the one given in [4, 14], where unimodal permutations are those whose inverse is left-unimodal in this paper, and in [17], where unimodal permutations are those whose inverse is right-unimodal in our terminology.

    • The permutation is unimodal.

      Claim 2.5.

      For , .

      Proof.

      A permutation is unimodal if either or its reversal is left-unimodal. The only permutations for which both and are left-unimodal are and . The formula now follows from Claim 2.2. ∎

      Remark 2.6.

      A permutation is unimodal if and only if

      for some . This happens if and only if , where is a hook with entries in the first column or in the first row, and is any hook with the same shape as . Thus unimodal permutations are a union of Knuth classes.

      2.2 Arc permutations

      Definition 2.7.

      A permutation is an arc permutation if, for every , the first letters in form an interval in (where the letter is identified with zero). Denote by the set of arc permutations in .

      • The permutation is an arc permutation in , but is not an arc permutation in , since is an interval in but not in .

        Claim 2.8.

        For , .

        Proof.

        To build , there are choices for and two choices for every other letter except the last one. ∎

        Remark 2.9.

        Arc permutations are not a union of Knuth classes. Note, however, that arc permutations may be characterized in terms of descent sets as follows. A permutation is an arc permutation if and only if

        for some .

        It is clear from the definition that the sets of left-unimodal, unimodal and arc permutations satisfy . We denote by the set of non-unimodal arc permutations. It follows from Remarks 2.6 and 2.9 that is not a union of Knuth classes. However, has some surprising Knuth-like properties, which will be described in Section 7.

        3 Characterization by pattern avoidance

        In this section the sets of left-unimodal permutations, arc permutations, and unimodal permutations are characterized in terms of pattern avoidance. Given a set of patterns , denote by the set of permutations in that avoid all of the , that is, that do not contain a subsequence whose entries are in the same relative order as those of . Define analogously.

        3.1 Left-unimodal permutations

        It will be convenient to use terminology from geometric grid classes. Studied by Albert et al. [5], a geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope , whose locations are determined by the positions of the corresponding entries in a matrix with entries in . More precisely, is the set of permutations that can be obtained by placing dots on the segments in such a way that there are no two dots on the same vertical or horizontal line, labeling the dots with by increasing -coordinate, and then reading them by increasing -coordinate. All the geometric grid classes that we consider in this paper are also profile classes in the sense of Murphy and Vatter [9].

        Left-unimodal permutations are those that can be drawn on the picture on the left of Figure 3.1, which consists of a segment of slope above a segment of slope . The picture on the right shows a drawing of the permutation . The grid class of permutations that can be drawn on this picture is denoted by

        so we have that

        Figure 3.1: The grid for left-unimodal permutations, and a drawing of the permutation .

        It is clear from the description that geometric grid classes are always closed under pattern containment, so they are characterized by the set of minimal forbidden patterns. In the case of left-unimodal permutations, we get the following description.

        Claim 3.1.

        .

        Proof.

        The condition that every prefix of is an interval in is equivalent to the condition that there is no pattern (with ) where the value of is between and , that is, avoids and . ∎

        3.2 Arc permutations

        Arc permutations can be characterized in terms of pattern avoidance, as those permutations avoiding the eight patterns with .

        Theorem 3.2.
        Proof.

        For an integer , denote by the element of that is congruent with mod . Let , and suppose that . Let be the smallest number with the property that is not an interval in . By minimality of , the set contains neither nor . Letting be such that , it follows that is an occurrence of one of the eight patterns above.

        Conversely, if contains one of the eight patterns, let be such an occurrence, where . Then is not an interval in .

        Corollary 3.3.

        for .

        Arc permutations can also be described in terms of grid classes, as those permutations that can be drawn on one of the two pictures in Figure 3.2. We write

        Figure 3.2: Grids for arc permutations.

        3.3 Unimodal permutations

        In terms of grid classes, unimodal permutations are those that can be drawn on one of the two pictures in Figure 3.3, that is,

        Figure 3.3: Grids for unimodal arc permutations.

        Next we characterize unimodal permutations in terms of pattern avoidance.

        Proposition 3.4.
        Proof.

        If contains or , then it is clear that is not unimodal. For the converse, we show that every arc permutation that is not unimodal must contain one of the patterns or . Since , it can be drawn on one of the two pictures in Fig. 3.2. Suppose it can be drawn on the left picture. Since is not unimodal, any drawing of on the left picture requires some element with to be on the first increasing slope, and some element with to be on the second increasing slope. Then is an occurrence of . An analogous argument shows that if can be drawn on the right picture in Fig. 3.2 but it is not unimodal, then it contains . ∎

        Corollary 3.5.

        for .

        4 Prefixes associated to the shifted staircase shape

        Consider the shifted staircase shape with rows labeled from top to bottom, and columns labeled from left to right. Given a filling with the numbers from to , with increasing entries in each row and column, erase the numbers greater than , for some , obtaining a partial filling of . For each of the remaining entries , if lies in row and column , let be the transposition . Associate to the partial filling the permutation , with multiplication from the right.

        • The partial filling

          missing
          1
          2
          3
          4
          5
          6
            missing2   3   4   5   6   7
          \young(12368,:45910,::7,:::,::::,:::::)

          corresponds to the product of transpositions

          Theorem 4.1.

          The set of permutations obtained as products of transpositions associated to a partial filling of the shifted staircase shape is exactly .

          Proof.

          The first observation is that if two boxes in the tableau are in different rows and columns, the associated transpositions commute. It follows that the resulting permutation depends only on what boxes of the tableaux are filled, but not on the order in which they were filled. For example, the partial filling

          1
          2
          3
          4
          5
          6
            2   3   4   5   6   7
          \young(12345,:6789,::10,:::,::::,:::::)

          yields again the permutation , just as the partial filling in the above example, since both have the same set of filled boxes.

          We claim that, from the set of filled boxes, the corresponding permutation can be read as follows. Let be the largest such that the box is filled. Then, starting at the bottom-left corner of that box, consider the path with north and east steps (along the edges of the boxes of the tableau) that separates the filled and unfilled boxes, ending at the top-right corner. At each east step, read the label of the corresponding column, and at each north step, read the label of the corresponding row. This claim can be easily proved by induction on the number of filled boxes. The permutations obtained by reading the labels of such paths are precisely the left-unimodal permutations. ∎

          The above proof gives a bijection between and the set of shifted shapes of size at most , which consist of the filled boxes in partial fillings.

          Definition 4.2.

          For , denote by the shifted shape corresponding to any partial filling of associated to .

          5 The weak order on

          5.1 A criterion for domination

          Let be the length function on the symmetric group with respect to the Coxeter generating set , where is identified with the adjacent transposition . Recall the definition of the (right) weak order on : for every pair , if and only if . Denote this poset by . Recall that is a lattice, which is not modular. First, we give a combinatorial criterion for weak domination of unimodal permutations.

          The concept of shifted shape from Definition 4.2 can be extended to all unimodal permutations as follows: for let , where denotes the longest permutation , which is the maximum in . Denote by the identity permutation, which is the minimum in . Note that .

          Theorem 5.1.

          For every pair , in if and only if

          • either or , and

          • .

          Proof.

          By [6, Cor. 1.5.2, Prop. 3.1.3], if in , then the corresponding descent sets satisfy . Combining this with the characterizations of left-unimodal and unimodal permutations by descent sets, given in Remarks 2.3 and 2.6, condition follows.

          Now we may assume, without loss of generality, that (for , the same proof holds by symmetry, by conjugation by ). To complete the proof it suffices to show that for two left-unimodal permutations, domination in weak order is equivalent to inclusion of the corresponding shapes. Indeed, recall the bijection from to the set of shifted shapes of size at most , described in Section 4. By this bijection, for any , the addition of a box in the border of corresponds to a switch of two adjacent increasing letters in giving a permutation in . This is precisely the covering relation in . Thus, for two left-unimodal permutations, the covering relation in is equivalent to the covering relation in the poset of shifted shapes inside ordered by inclusion, and hence domination is equivalent. ∎

          Corollary 5.2.

          For every

          where denotes the size of the shape.

          5.2 Enumeration of maximal chains

          Denote by the subposet of which is induced by . Theorem 5.1 implies the following nice properties of this poset.

          Corollary 5.3.

          is a graded self-dual modular lattice.

          Corollary 5.4.

          For every , the number of maximal chains in the interval is equal to the number of standard Young tableaux of shifted shape , hence given by a hook formula.

          Proof.

          By Theorem 4.1 together with Theorem 5.1, the statement holds for every . By conjugation by , it holds for all elements in as well. ∎

          Corollary 5.5.

          For , the number of maximal chains in is equal to twice the number of standard Young tableaux of shifted staircase shape, hence equal to

          Proof.

          The maximum covers the two elements and . Thus the number of maximal chains in is the sum of the numbers of maximal chains in and . By Corollary 5.4, this equals the number of standard Young tableaux of shape plus number of standard Young tableaux of shape . Since , these two shapes are the same, namely with the box in row (the bottommost row) removed. By Schur’s Formula [15][8, p. 267 (2)], the number of standard Young tableaux of this shape is , completing the proof. ∎

          5.3 The Hasse diagram

          Let be the undirected Hasse diagram of . A drawing of is given by the black vertices and solid edges in Figure 6.1.

          Proposition 5.6.
          • The diameter of is .

          • The vertices and are antipodes in .

          • The number of geodesics between and is .

          Proof.

          Since is a modular lattice, the distance between any two vertices is equal to the difference between the ranks of their join and their meet (see [1, Lemma 5.2]). Hence, the diameter is equal to the maximum rank. This proves and . Part then follows from Corollary 5.5. ∎

          6 A graph structure on arc permutations

          6.1 The graph

          Let be the subgraph of the Cayley graph induced by . In other words, the vertex set of is , and two elements are adjacent if and only if there exists a simple reflection , such that . The graph is drawn in Figure 6.1. The following theorem shows that and share similar properties.

          Theorem 6.1.
          • The diameter of is .

          • The vertices and are antipodes in .

          • The number of vertices in geodesics between and is .

          • The number of geodesics between and is .

          This theorem will be proved Subsections 6.2 and 6.3.

          Figure 6.1: The graph . The vertices not lying in a geodesic between and are drawn in red with dotted edges, and they correspond to non-unimodal permutations by Lemma 6.4.

          6.2 The diameter of

          In this subsection we show that the diameter of is , proving Theorem 6.1(i). To see that this is a lower bound, note that the inversion number does not change by more than 1 along each edge of . It follows that the diameter of is at least . This argument also shows that part (ii) of Theorem 6.1 will follow once we prove part , since the distance between and is at least .

          The proof that this is also an upper bound on the diameter is more involved, and it is similar to the proof in [1, Theorem 5.1]. Consider the encoding given by , where

          and, for ,

          where denotes the element of that is congruent with mod . Note that exactly one of the two above conditions holds, because forms an interval in .

          • For , .

            The encoding of the vertices of is given in Figure 6.2. The following observation is clear from the definition of and the encoding .

            Lemma 6.2.

            Two arc permutations with are adjacent in if and only if exactly one of the following holds:

            • is obtained from by switching two adjacent entries and for some ;

            • for all ;

            • mod , and for all .

            Figure 6.2: The graph with its vertices encoded by (commas and parenthesis have been removed). Deleting the two dotted blue edges gives the undirected Hasse diagram of the dominance order on .

            The set of possible encodings inherits the dominance order from , that is, if and only if for every ,

            The covering relations in this poset are almost identical to those described by Lemma 6.2. More precisely, we have the following result.

            Proposition 6.3.

            Through the encoding , the graph is isomorphic to the undirected Hasse diagram of the dominance order on with the additional edges arising from Lemma 6.2(iii) with .

            Denote by the distance function in the undirected Hasse diagram of the dominance order. To compute , let us first recall some basic facts. The dominance order on is a ranked poset where

            This poset is a modular lattice, with

            where for every , and

            where for every . Finally, recall that the distance between two elements in the undirected Hasse diagram of a modular lattice is equal to the difference between the ranks of their join and their meet, see e.g. [1, Lemma 5.2].

            Combining these facts implies that

            (1)

            Now we are ready to prove the upper bound on diameter of . Denoting by the distance function in , we will show that for any , .

            Let be the -cycle . Clearly, is invariant under left multiplication by . Moreover, left multiplication by is an automorphism of . Thus, for any integer ,

            (2)

            where the last inequality follows from Proposition 6.3. Let . By equation (1),

            (3)

            where for . Note that for every . If , then

            and we are done.

            Otherwise, we can assume without loss of generality that . Let , so that . Note that for , we have and . Thus, by equation (1),

            Combining this formula with equations (2) and (3), we get

            If for all , then

            Otherwise, since for all , there must be some such that or . If for some , then for all , so . Similarly, if for some , then , completing the proof of Theorem 6.1(i).

            6.3 Geodesics of

            To prove parts (ii) and (iii) of Theorem 6.1 we need the following lemma.

            Lemma 6.4.

            A permutation in lies in a geodesic between and if and only if it is unimodal.

            Proof.

            By Corollary 5.3, all unimodal permutations lie in geodesics between and in the undirected Hasse diagram of . By Proposition 5.6 and Theorem 6.1(i), the distance between and in this Hasse diagram is the same as in , thus the geodesics between these vertices in this Hasse diagram are also geodesics in .

            It remains to show that for every non-unimodal arc permutations , is not in a geodesic between and . It suffices to prove that for every such , either , or . These two cases are analogous to the dichotomy in Remark 2.9 and Figure 3.2.

            If , then , since otherwise would be unimodal. Let , and suppose for contradiction that . Then there is a sequence of arc permutations where each is obtained from by switching two adjacent letters at a descent, decreasing the number of inversions by one. In particular, in every , the entry is to the left of , and is to the right of . In order to remove the inversion created by the pair in we would have to switch and , which would create a permutation containing , thus not in by Theorem 3.2. This shows that .

            Similarly, if , then . Let , and suppose for contradiction that . Then there is a sequence of arc permutations where each is obtained from by switching two adjacent letters at an ascent, increasing the number of inversions by one. Again, this is impossible because after switching the pair , the entries would form an occurrence of , so the permutation would not be in by Theorem 3.2. We conclude that . ∎

            Proof of Theorem 6.1.

            Parts and were proved in Subsection 6.2. To prove (iii), combine Lemma 6.4 with Claim 2.5. Finally, (iv) follows from Lemma 6.4 together with Corollary 5.5. ∎

            7 Equidistribution

            In this section we show that the descent set is equidistributed on arc permutations that are not unimodal and on the set of standard Young tableaux obtained from hooks by adding one box in position .

            7.1 Enumeration of arc permutations by descent set

            For a set , define .

            Proposition 7.1.

            For ,

            Proof.

            Let , and let .

            If , then can be drawn on the picture on the left of Figure 3.3. The generating function for these permutations with respect to the descent set is