Egge and Mansour have recently studied permutations which avoid 1243 and 2143 regarding the occurrence of certain additional patterns. Some of the open questions related to their work can easily be answered by using permutation diagrams. Like for 132-avoiding permutations the diagram approach gives insights into the structure of {1243,2143}-avoiding permutations that yield simple proofs for some enumerative results concerning forbidden patterns in such permutations.
We analyze the structure and enumerate Dumont permutations of the first and second kinds avoiding certain patterns or sets of patterns of length 3 and 4. Some cardinalities are given by Catalan numbers, powers of 2, little Schroeder numbers, and other known or related sequences.
The diagram of a 132-avoiding permutation can easily be characterized: it is simply the diagram of a partition. Based on this fact, we present a new bijection between 132-avoiding and 321-avoiding permutations. We will show that this bijection translates the correspondences between these permutations and Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively, to each other. Moreover, the diagram approach yields simple proofs for some enumerative results concerning forbidden patterns in 132-avoiding permutations.
In this paper we study a mapping from permutations to Dyck paths. A Dyck path gives rise to a (Young) diagram and we give relationships between statistics on permutations and statistics on their corresponding diagrams. The distribution of the size of this diagram is discussed and a generalisation given of a parity result due to Simion and Schmidt. We propose a filling of the diagram which determines the permutation uniquely. Diagram containment on a restricted class of permutations is shown to be related to the strong Bruhat poset.
We generalize the work of Fomin, Greene, Reiner, and Shimozono on balanced labellings in two directions: (1) we define the diagrams of affine permutations and the balanced labellings on them; (2) we define the set-valued version of the balanced labellings. We show that the column-strict balanced labellings on the diagram of an affine permutation yield the affine Stanley symmetric function defined by Lam, and that the column-strict set-valued balanced labellings yield the affine stable Grothendieck polynomial of Lam. Moreover, once we impose suitable flag conditions, the flagged column-strict s…
A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths. The aim of this article is to extend the notion of local types from permutations to $k$-Stirling permutations (also known as $k$-multipermutati…
The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213- and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.
We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young t…
We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of $[n] \times [n]$ to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over …
Generalizing the notion of a vexillary permutation, we introduce a filtration of S_infinity by the number of Schur function terms in the Stanley symmetric function, with the kth filtration level called the k-vexillary permutations. We show that for each k, the k-vexillary permutations are characterized by avoiding a finite set of patterns. A key step is the construction of a Specht series, in the sense of James and Peel, for the Specht module associated to the diagram of a permutation. As a corollary, we prove a conjecture of Liu on diagram varieties for certain classes of permutation diagrams…
There are numerous combinatorial objects associated to a Grassmannian permutation $w_λ$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations $w$ that are not necessarily Grassmannian. We give two main results: first, we show that certain acyclic orientations, rook placements avoiding a diagram of $w$, and fillings of a diagram of $w$ are equinumerous for all permutations $w$. Second, we give a $q$-analogue of a result of Hultman-Linusson-Shareshian-Sjöstrand by showing that under a certain pattern condit…
Given two operators $\hat D$ and $\hat E$ subject to the relation $\hat D\hat E -q \hat E \hat D =p$, and a word $w$ in $M$ and $N$, the rewriting of $w$ in normal form is combinatorially described by rook placements in a Young diagram. We give enumerative results about these rook placements, particularly in the case where $p=(1-q)/q^2$. This case naturally arises in the context of the PASEP, a random process whose partition function and stationary distribution are expressed using two operators $D$ and $E$ subject to the relation $DE-qED=D+E$ (matrix Ansatz). Using the link obtained by Corteel…
Diaconis and Graham studied a measure of distance from the identity in the symmetric group called total displacement and showed that it is bounded below by the sum of length and reflection length. They asked for a characterization of the permutations where this bound is an equality; we call these the shallow permutations. Cornwell and McNew recently interpreted the cycle diagram of a permutation as a knot diagram and studied the set of permutations for which the corresponding link is an unlink. We show the shallow permutations are precisely the unlinked permutations. As Cornwell and McNew give…
For a permutation $π$ in the symmetric group $S_n$ let the {\it total degree} be its valency in the Hasse diagram of the strong Bruhat order on $S_n$, and let the {\it down degree} be the number of permutations which are covered by $π$ in the strong Bruhat order. The maxima of the total degree and the down degree and their values at a random permutation are computed. Proofs involve variants of a classical theorem of Turán from extremal graph theory.
We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.
A permutation of size $n$ can be identified to its diagram in which there is exactly one point per row and column in the grid $[n]^2$. In this paper we consider multidimensional permutations (or $d$-permutations), which are identified to their diagrams on the grid $[n]^d$ in which there is exactly one point per hyperplane $x_i=j$ for $i\in[d]$ and $j\in[n]$. We first investigate exhaustively all small pattern avoiding classes. We provide some bijection to enumerate some of these classes and we propose some conjectures for others. We then give a generalization of well-studied Baxter permutation…
In their paper [1] on Wilf-equivalence for singleton classes, Backelin, Xin, and West introduce a transformation $ϕ^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. In [3], Bousquet-M$\acute{\textrm{e}}$lou and Steingr$\acute{\textrmı}$msson prove the analogue of the main result of [1] in the context of involutions, and in so doing they must prove that $ϕ^*$ commutes with the operation of taking inverses. The proof of this commutation result is long and difficult, and Bousquet-M$\acute{\textrm{e}}$lou and Steingr$\acute{\textrmı}$msson ask if $…
We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries these numbers of matrices are not polynomials in q (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even pol…
Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which…
The graph of zigzag diagrams is a close relative of Young's lattice. The boundary problem for this graph amounts to describing coherent random permutations with descent-set statistic, and is also related to certain positive characters on the algebra of quasi-symmetric functions. We establish connections to some further relatives of Young's lattice and solve the boundary problem by reducing it to the classification of spreadable total orders on integers, as recently obtained by Jacka and Warren.
Noting that cycle diagrams of permutations visually resemble grid diagrams used to depict knots and links in topology, we consider the knot (or link) obtained from the cycle diagram of a permutation. We show that the permutations which correspond in this way to an unknot are enumerated by the Schröder numbers, and also enumerate the permutations corresponding to an unlink. The proof uses Bennequin's inequality.
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, then it becomes an arc diagram. The count of arc diagrams is known to have a rich structure. We show that the count of polygon diagrams exhibits the same interesting behaviours, in particular it is almost polynomial in the number of points on the boundary components, and the leadi…
The research on pattern-avoidance has yielded so far limited knowledge on Wilf-ordering of permutations. The Stanley-Wilf limits sqrt[n](|S_n(tau)|) and further works suggest asymptotic ordering of layered versus monotone patterns. Yet, Bona has provided the only known up to now result of its type on ordering of permutations: |S_n(1342)|<|S_n(1234)|<|S_n(1324)| for n>6. We give a different proof of this result by ordering S_3 up to the stronger shape-Wilf-order: |S_Y(213)|<=|S_Y(123)|<=|S_Y(312)| for any Young diagram Y, derive as a consequence that |S_Y(k+2,k+1,k+3,tau)|<=|S_Y(k+1,k+2,k+3,tau…
We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns t_1...t_{m-2}m(m-1) and t_1...t_{m-2}(m-1)m in a permutation, respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As special case we derive a one-to-one correspondence between permutations which avoid each of the patterns t_1...t_{m-2}m(m-1) in S_m and such ones which avoid each of the patterns t_1...t_{m-2}(m-1)m. For m=3, this correspondence coincides with the bijection given by S…
Rothe diagrams are diagrams which track inversions of a permutation. We define six main properties that Rothe diagrams fulfill: the southwest, dot, popping, numbering, step-out avoiding, and empty cell gap rules. We prove that -- given an arbitrary bubble diagram -- four different subsets of these properties provide sufficient criteria for the diagram to be a Rothe diagram. We also prove that when a set of ordered, freely floating, non-empty columns satisfy the numbering and step-out avoiding rules, then they can be arranged into a Rothe diagram.
Starting from some considerations we make about the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that these (so-called bi-increasing) permutations are just the 321-avoiding ones. The paper investigates their excedance and descent structure. In particular, we find some nice combinatorial interpretations for the distribution coefficients of the number of excedances and descents, respectively, and their difference analogues over the bi-increasing permutations in…
In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the Le-diagrams of Alex Postnikov. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative si…
We study the shape of the Young diagram λassociated via the Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n\to\infty the rows of λbehave like the eigenvalues of a certain random matrix (traceless Gaussian Unitary Ensemble) with d rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asy…
Edelman and Greene constructed a correspondence between reduced words of the reverse permutation and standard Young tableaux. We prove that for any reduced word the shape of the region of the insertion tableau containing the smallest possible entries evolves exactly as the upper-left component of the permutation's (Rothe) diagram. Properties of the Edelman-Greene bijection restricted to 132-avoiding and 2143-avoiding permutations are presented. We also consider the Edelman-Greene bijection applied to non-reduced words.
The notion of set-valued Young tableaux was introduced by Buch in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson, Miller and Yong showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram $D(w)$ of a permutation $w$, a set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the rows are weakly d…
The pattern-avoiding fillings of Young diagrams we study arose from Postnikov's work on positive Grassman cells. They are called Le-diagrams, and are in bijection with decorated permutations. Other closely-related diagrams are interpreted as acyclic orientations of some bipartite graphs. The definition of the diagrams is the same but the avoided patterns are different. We give here bijections proving that the number of pattern-avoiding filling of a Young diagram is the same, for these two different sets of patterns. The result was obtained by Postnikov via a reccurence relation. This relation …
We study asymptotics of an irreducible representation of the symmetric group S_n corresponding to a balanced Young diagram λ(a Young diagram with at most C\sqrt{n} rows and columns for some fixed constant C) in the limit as n tends to infinity. We show that there exists a constant D (which depends only on C) with a property that |χ^λ(π)| = | Tr ρ^λ(π)/Tr ρ^λ(e) | < [ D max(1,|π|^2/n) / \sqrt{n}} ]^{|π|}, where |π| denotes the length of a permutation (the minimal number of factors necessary to write πas a product of transpositions). Our main tool is an analogue of Frobenius character formula wh…
Kohnert proposed the first monomial positive formula for Schubert polynomials as the generating polynomial for certain unit cell diagrams obtained from the Rothe diagram of a permutation. Billey, Jockusch and Stanley gave the first proven formula for Schubert polynomials as the generating polynomial for compatible sequences of reduced words of a permutation. In this paper, we give an explicit bijection between these two models, thereby definitively proving Kohnert's rule for Schubert polynomials.
Light-cone string diagrams have been used to reproduce the orbifold Euler characteristic of moduli spaces of punctured Riemann surfaces at low genus and with few punctures. Nakamura studied the meromorphic differential introduced by Giddings and Wolpert to characterise light-cone diagrams and introduced a class of graphs related to this differential. These Nakamura graphs were used to parametrise the cells in a light-cone cell decomposition of moduli space. We develop links between Nakamura graphs and realisations of the worldsheet as branched covers. This leads to a development of the combina…
Permutations and associated algebras allow the construction of half and quarter BPS operators in maximally supersymmetric Yang Mills theory with $U(N)$, $SO(N)$ and $Sp(N) $ gauge groups. The construction leads to bases for the operators, labelled by Young diagrams and associated group theory data, which have been shown to be orthogonal under the inner product defined by the free field two-point functions. In this paper, we study in detail the orientifold projection map between the Young diagram basis for $U(N)$ theories and the Young diagram basis for $ SO(N)$ (and $ Sp(N)$) half-BPS operator…
