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In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in $\mathbb{R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements., Comment: 45 pages, 13 figures

In this article, we study the enumeration by length of several walk models on the square lattice. We obtain bijections between walks in the upper half-plane returning to the $x$-axis and walks in the quarter plane. A recent work by Bostan, Chyzak, and Mahboubi has given a bijection for models using small north, west, and south-east steps. We adapt and generalize it to a bijection between half-plane walks using those three steps in two colours and a quarter-plane model over the symmetrized step set consisting of north, north-west, west, south, south-east, and east. We then generalize our bijections to certain models with large steps: for given $p\geq1$, a bijection is given between the half-plane and quarter-plane models obtained by keeping the small south-east step and replacing the two steps north and west of length 1 by the $p+1$ steps of length $p$ in directions between north and west. This model is close to, but distinct from, the model of generalized tandem walks studied by Bousquet-Mélou, Fusy, and Raschel.

R. Sulzgruber's rim hook insertion and the Hillman–Grassl correspondence are two distinct bijections between the reverse plane partitions of a fixed partition shape and multisets of rim-hooks of the same partition shape. It is known that Hillman–Grassl may be equivalently defined using the Robinson–Schensted–Knuth correspondence, and we show the analogous result for Sulzgruber's insertion. We refer to our description of Sulzgruber's insertion as diagonal RSK. As a consequence of this equivalence, we show that Sulzgruber's map from multisets of rim hooks to reverse plane partitions can be expressed in terms of Greene–Kleitman invariants.

We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields can be converted to type (i) compositions, whereas Carlitz and "locally Mullen" compositions can be formulated as type (ii) compositions. We use the multisection formula to translate the problem from integers to group elements, the transfer matrix method to do exact counting, and finally the Perron-Frobenius theorem to derive asymptotics. We also exhibit bijections involving certain restricted classes of compositions.

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an efficient scheme for encoding simple outerplanar maps.

Restricted growth functions (RGFs) avoiding the pattern $1212$ are in natural bijection with noncrossing partitions. Motivated by recent work of Campbell et al., we study five classical statistics bk, ls, lb, rs and rb on $1212$-avoiding RGFs. We show the equidistribution of (ls, rb, lb, bk) and (rb, ls, lb, bk) on $1212$-avoiding RGFs by constructing a simple involution. To our surprise, this result was already proved by Simion 22 years ago via an involution on noncrossing partitions. Our involution, though turns out essentially the same as Simion's, is defined quite differently and has the advantage that makes the discussion more transparent. Consequently, a multiset-valued extension of Simion's result is discovered. Furthermore, similar approach enables us to prove the equidistribution of (mak, rb, rs, bk) and (rb, mak, rs, bk) on $1212$-avoiding RGFs, where "mak" is a set partition statistic introduced by Steingrímsson.Through two bijections to Motzkin paths, we also prove that the triple of classical permutation statistics (exc+1, den, inv — exc) on $321$-avoiding permutations is equidistributed with the triple (bk, rb, rs) on $1212$-avoiding RGFs, which generalizes another result of Simion. In the course, an interesting $q$-analog of the $\gamma$-positivity of Narayana polynomials is found.

We consider the classical Mahonian statistics on the set $B_n(\Sigma)$ of signed permutations in the hyperoctahedral group $B_n$ which avoid all patterns in $\Sigma$, where $\Sigma$ is a set of patterns of length two. In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\left|B_n(\Sigma)\right|$ is constant whenever $\left|\Sigma\right|=1$, whereas in most but not all instances where $\left|\Sigma\right|=2$, $\left|B_n(\Sigma)\right|=(n+1)!$. We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\left|B_n(\Sigma)\right|=(n+1)!$. In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on $B_n(21, \bar{2}\bar{1})$ and by giving the major index on $D_n(\Sigma)$ for $\Sigma =\{21, \bar{2}\bar{1}\}$ and $\Sigma=\{12,21\}$. The main result of this paper is to give the inversion generating functions for $B_n(\Sigma)$ for almost all sets $\Sigma$ with $\left|\Sigma\right|\leq2.$

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations., Comment: 23 pages. formerly part of arXiv:1305.1314

As a variation of De Bruijn graphs on strings of symbols, the graph of overlapping permutations has a directed edge $\pi(1)\pi(2)\ldots \pi(n+1)$ from the standardization of $\pi(1)\pi(2)\ldots \pi(n)$ to the standardization of $\pi(2)\pi(3)\ldots \pi(n+1)$. In this paper, we consider the enumeration of $d$-cycles in the subgraph of overlapping $(231, 4\bar{1}32)$-avoiding permutations. To this end, we introduce the notions of marked Motzkin paths and marked Riordan paths, where a marked Motzkin (resp. Riordan) path is a Motzkin (resp. Riordan) path in which exactly one step before the leftmost return point is marked. We show that the number of closed walks of length $d$ in the subgraph of overlapping $(231, 4\bar{1}32)$-avoiding permutations are closely related to the number of marked Motzkin paths and that of marked Riordan paths. By establishing bijections, we get the enumerations of marked Motzkin paths and marked Riordan paths. As a corollary, we provide bijective proofs of two identities involving Catalan numbers in answer to the problem posed by Ehrenborg, Kitaev and Steingrímsson. Moreover, we get the enumerations of $(231, 4\bar{1}32)$-avoiding affine permutations and $(312, 32\bar{4}1)$-avoiding affine permutations.

Centrosymmetric involutions in the symmetric group ${\mathcal S}_{2n}$ are permutations $\pi$ such that $\pi=\pi^{-1}$ and $\pi(i)+\pi(2n+1-i)=2n+1$ for all $i$ , and they are in bijection with involutions of the hyperoctahedral group. We describe the distribution of some natural descent statistics on $321$ -avoiding centrosymmetric involutions , including the number of descents in the first half of the involution, and the sum of the positions of these descents. Our results are based on two new bijections , one between centrosymmetric involutions in $ {\mathcal S} _{2n}$ and subsets of $\{1,\dots,n\}$ , and another one showing that certain statistics on Young diagrams that fit inside a rectangle are equidistributed . We also use the latter bijection to refine a known result stating that the distribution of the major index on $321$ -avoiding involutions is given by the $q$ -analogue of the central binomial coefficients.

A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with a precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Finally, we share some the results of a computer program developed to calculate tableaux inversions., Comment: 20 pages, corrected typos

Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrimsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdos-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound., Comment: 29 pages, 3 tables, 1 figure

We investigate pattern avoidance in alternating permutations and an alternating analogue of Young diagrams. In particular, using an extension of Babson and West's notion of shape- Wilf equivalence described in our recent paper (with N. Gowravaram ), we generalize results of Backelin , West, and Xin and Ouchterlony to alternating permutations. Unlike Ouchterlony and Bona 's bijections , our bijections are not the restrictions of Backelin , West, and Xin's bijections to alternating permutations. This paper is the second of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (with N. Gowravaram , arXiv: 1301. 6796v1 ). The first paper in the series is Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (with N. Gowravaram , Electronic Journal of Combinatorics 20(4):# P17 , 2013).

In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux which gives a clear proof that tableaux of size n are counted by n!, and which moreover respects most of the well-known statistics studied originally on alternative and permutation tableaux. Our insertion procedure allows to define in particular two simple new bijections between tree-like tableaux and permutations: the first one is conceived specifically to respect the generalized pattern 2-31, while the second one respects the underlying tree of a tree-like tableau., Dans ce travail nous introduisons et étudions les tableaux boisés, qui sont certains remplissages de diagrammes de Ferrers en bijection simple avec les tableaux de permutation et les tableaux alternatifs. Nous décrivons une procédure d'insertion élémentaire sur nos tableaux qui donne une preuve limpide que les tableaux de taille n sont comptés par n!, et qui de plus respecte la plupart des statistiques standard sur les tableaux de permutation et tableaux alternatifs. Notre procédure d'insertion permet en particulier de définir deux nouvelles bijections simples entre tableaux et permutations: la première est conçue spécifiquement pour respecter le motif généralisé 2-31 sur les permutations, tandis que la deuxième respecte l'arbre binaire sous-jacent à un tableau boisé.

In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and that of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, the map which we define bijects facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid\left\langle v,\alpha \right\rangle=m\}$ as separating wall. As a result, it restricts to a bijection between the set of facets of the positive part of $\Delta^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}^m(A_n)$. Our map is a composition of two bijections in which integer partitions in an $m$-dilated $n$-staircase shape come into play.

The number of regions of the type $C_n$ Shi arrangement in $\mathbb{R}^n$ is $(2n+1)^n$. Strikingly, no bijective proof of this fact has been given thus far. The aim of this paper is to provide such a bijection and use it to prove more refined results. We construct a bijection between the regions of the type $C_n$ Shi arrangement in $\mathbb{R}^n$ and sequences $a_1a_2 \ldots a_n$, where $a_i \in \{-n, -n+1, \ldots, -1, 0, 1, \ldots, n-1, n\}$, $ i \in [n]$. Our bijection naturally restrict to bijections between special regions of the arrangement and sequences with a given number of distinct elements.

We give bijective proofs that, when combined with one of the combinatorial proofs of the general ballot formula, constitute a combinatorial argument yielding the number of lattice paths from $(0,0)$ to $(n,rn)$ that touch or cross the diagonal $y = rx$ at exactly $k$ lattice points. This enumeration partitions all lattice paths from $(0,0)$ to $(n,rn)$. While the resulting formula can be derived using results from Niederhausen, the bijections and combinatorial proof are new.

We obtain the reduced form of the “combinatorial logarithm” Ω by looking at bijections related to connected point-determining cographs and connected co-point-determining graphs.

We identify several subsets of the partitions of $[n]$, each characterized by the avoidance of a pair of patterns, respectively of lengths four and five. Each of the classes we consider is enumerated by the Catalan numbers. Furthermore, the members of each class having a prescribed number of blocks are enumerated by the Narayana numbers. We use both algebraic and combinatorial methods to establish our results. In some of the cases, we make use of the kernel method to solve the recurrence arising when a further statistic is considered. In other cases, we define bijections with previously enumerated classes which preserve the number of blocks. Two of our bijections are of an algorithmic nature and systematically replace the occurrences of one pattern with those of another having the same length.

A strong complete mapping of a group $G$ is a bijection $\theta\colon G\to G$ for which both mappings $x\mapsto x\theta(x)$ and $x\mapsto x^{-1}\theta(x)$ are bijections. We characterize finite abelian groups that admit strong complete mappings, thus solving a problem posed by Horton in 1990. We also prove the existence of strong complete mappings for countably infinite groups. A corrigendum for this paper was added on 2 October 2018.

Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a single square partition or with several partitions with only one part. The cardinalities of those families of partitions are the Catalan and ballot numbers, respectively. In this paper we present a non-recursive description for those families and prove that the generating function of the size of those partitions is a Kostka number. We also present bijections between those sets of partitions and sets of trees and forests enumerated by the Catalan an ballot numbers, respectively.

The classical Chung-Feller Theorem offers an elegant perspective for enumerating the Catalan number $c_n= \frac{1}{n+1}\binom{2n}{n}$. One of the various proofs is by the uniform-partition method. The method shows that the set of the free Dyck $n$-paths, which have $\binom{2n}{n}$ in total, is uniformly partitioned into $n+1$ blocks, and the ordinary Dyck $n$-paths form one of these blocks; therefore the cardinality of each block is $\frac{1}{n+1}\binom{2n}{n}$. In this article, we study the Chung-Feller property: a sup-structure set can be uniformly partitioned such that one of the partition blocks is (isomorphic to) a well-known structure set. The previous works about the uniform-partition method used bijections, but here we apply generating functions as a new approach. By claiming a functional equation involving the generating functions of sup- and sub-structure sets, we re-prove two known results about Chung-Feller property, and explore several new examples including the ones for the large and the little Schröder paths. Especially for the Schröder paths, we are led by the new approach straightforwardly to consider "weighted" free Schröder paths as sup-structures. The weighted structures are not obvious via bijections or other methods.

(Dual-)promotion and (dual-)evacuation are bijections on SYT(\lambda) for any partition \lambda. Let c^r denote the rectangular partition (c,...,c) of height r, and let sc_k (k > 2) denote the staircase partition (k,k-1,...,1). B. Rhoades showed representation-theoretically that promotion on SYT(c^r) exhibits the cyclic sieving phenomenon (CSP). In this paper, we demonstrate a promotion- and evacuation-preserving embedding of SYT(sc_k) into SYT(k^{k+1}). This arose from an attempt to demonstrate the CSP of promotion action on SYT(sc_k)., Comment: 14 pages, typos corrected

This paper deals with several bijections between cladograms and perfect matchings. The first of these is due to Diaconis and Holmes. The second is a modification of the Diaconis-Holmes matching which makes deletion of the largest labeled leaf correspond to gluing together the last two points in the perfect matching. The third is an entirely new encoding of cladograms, first as a bijection with a certain set of strings and then via this to perfect matchings. In this pair of bijections, deletion of the largest labeled leaf corresponds to deletion of the corresponding symbols from the string or deletion of the corresponding pair from the matching. These two new bijections are related through a common max-min labeling of internal vertices with two different choices for the label of the root node. All these encodings are extended to cladograms with edge lengths and left-right ordered children. Moving a single symbol in this last encoding corresponds to a subtree prune and regraft operation on the cladogram, making it well suited for use in phylogentics software. Finally, a perfect Gray code for cladograms is derived from the bar encoding, along with a total ordering on all cladograms, Algorithms are also provided for finding the next and previous cladogram, the cladogram at any position, and the position of any cladogram in the sequence.

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.

We construct an involution on set partitions which keeps track of the numbers of crossings, nestings and alignments of two edges. We derive then the symmetric distribution of the numbers of crossings and nestings in partitions, which generalizes Klazar's recent result in perfect matchings. By factorizing our involution through bijections between set partitions and some path diagrams we obtain the continued fraction expansions of the corresponding ordinary generating functions., Comment: 12 pages

$S_n(\pi_1,\pi_2,\dots, \pi_r)$ denotes the set of permutations of length $n$ that have no subsequence with the same order relations as any of the $\pi_i$. In this paper we show that $|S_n(1342,2143)|=|S_n(3142,2341)|$ and $|S_n(1342,3124)|=|S_n(1243,2134)|$. These two facts complete the classification of Wilf-equivalence classes for pairs of permutations of length four. In both instances we exhibit bijections between the sets using the idea of a "block", and in the former we find a generating function for $|S_n(1342,2143)|$.

Given two positive integers $m\le n$, we consider the set of partitions $\lambda=(\lambda_1,\dots,\lambda_\ell,0,\dots)$, $\lambda_1\ge\lambda_2\ge\dots$, of $n$ such that the sum of its parts over a fixed increasing subsequence $(a_j)$ is $m$: $\lambda_{a_1}+\lambda_{a_2}+\dots=m$. We show that the number of such partitions does not depend on $n$ if $m$ is either constant and small relatively to $n$ or depend on $n$ but is close to its largest possible value: $n-ma_1=k$ for fixed $k$ (in the latter case some additional requirements on the sequence $(a_j)$ are needed). This number is equal to the number of so-called colored partitions of $m$ (respectively $k$). It is proved by constructing bijections between these objects.

This article describes conjectured combinatorial interpretations for the higher $q,t$-Catalan sequences introduced by Garsia and Haiman, which arise in the theory of symmetric functions and Macdonald polynomials. We define new combinatorial statistics generalizing those proposed by Haglund and Haiman for the original $q,t$-Catalan sequence. We prove explicit summation formulas, bijections, and recursions involving the new statistics. We show that specializations of the combinatorial sequences obtained by setting $t=1$ or $q=1$ or $t=1/q$ agree with the corresponding specializations of the Garsia-Haiman sequences. A third statistic occurs naturally in the combinatorial setting, leading to the introduction of $q,t,r$-Catalan sequences. Similar combinatorial results are proved for these trivariate sequences.

Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the Garsia-Haiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules.

In 1958, Richard Guy proved that the number of partitions of $n$ into odd parts greater than one equals the number of partitions of $n$ into distinct parts with no powers of 2 allowed, which is closely related to Euler's famous theorem that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We consider extensions of Guy's result, which naturally lead to a new algorithm for producing bijections between various equivalent partition ideals of order 1, as well as to two new infinite families of parity results which follow from Euler's Pentagonal Number Theorem and a well-known series-product identity of Jacobi.

We generalize the notion of pattern avoidance to arbitrary functions on ordered sets, and consider specifically three scenarios for permutations: linear, cyclic and hybrid, the first one corresponding to classical permutation avoidance. The cyclic modification allows for circular shifts in the entries. Using two bijections, both ascribable to both Deutsch and Krattenthaler independently, we single out two geometrically significant classes of Dyck paths that correspond to two instances of simultaneous avoidance in the purely linear case, and to two distinct patterns in the hybrid case: non-decreasing Dyck paths (first considered by Barcucci et al.), and Dyck paths with at most one long vertical or horizontal edge. We derive a generating function counting Dyck paths by their number of low and high peaks, long horizontal and vertical edges, and what we call sinking steps. This translates into the joint distribution of fixed points, excedances, deficiencies, descents and inverse descents over 321-avoiding permutations. In particular we give an explicit formula for the number of 321-avoiding permutations with precisely $k$ descents, a problem recently brought up by Reifegerste. In both the hybrid and purely cyclic scenarios, we deal with the avoidance enumeration problem for all patterns of length up to 4. Simple Dyck paths also have a connection to the purely cyclic case; here the orbit-counting lemma gives a formula involving the Euler totient function and leads us to consider an interesting subgroup of the symmetric group.

Let $d(n)$ count the lattice paths from $(0,0)$ to $(n,n)$ using the steps (0,1), (1,0), and (1,1). Let $e(n)$ count the lattice paths from $(0,0)$ to $(n,n)$ with permitted steps from the step set ${\bf N} \times {\bf N} - \{(0,0)\}$, where ${\bf N}$ denotes the nonnegative integers. We give a bijective proof of the identity $e(n) = 2^{n-1} d(n)$ for $n \ge 1$. In giving perspective for our proof, we consider bijections between sets of lattice paths defined on various sets of permitted steps which yield path counts related to the Narayana polynomials.

We define and study $m$-closed cellular algebras (coherent configurations) and $m$-isomorphisms of cellular algebras which can be regarded as $m$th approximations of Schurian algebras (i.e. the centralizer algebras of permutation groups) and of strong isomorphisms (i.e. bijections of the point sets taking one algebra to the other) respectively. If $m=1$ we come to arbitrary cellular algebras and their weak isomorphisms (i.e. matrix algebra isomorphisms preserving the Hadamard multiplication). On the other hand, the algebras which are $m$-closed for all $m\ge 1$ are exactly Schurian ones whereas the weak isomorphisms which are $m$-isomorphisms for all $m\ge 1$ are exactly ones induced by strong isomorphisms. We show that for any $m$ there exist $m$-closed algebras on $O(m)$ points which are not Schurian and $m$-isomorphisms of cellular algebras on $O(m)$ points which are not induced by strong isomorphisms. This enables us to find for any $m$ an edge colored graph with $O(m)$ vertices satisfying the $m$-vertex condition and having non-Schurian adjacency algebra. On the other hand, we rediscover and explain from the algebraic point of view the Cai-Fürer-Immerman phenomenon that the $m$-dimensional Weisfeiler-Lehman method fails to recognize the isomorphism of graphs in an efficient way.