Your data matches 251 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
St000972: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 2
([],3)
=> 1
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 4
([(0,1),(0,2),(1,2)],3)
=> 5
Description
The composition number of a graph. This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.
St001474: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 2
([],3)
=> 1
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 4
([(0,1),(0,2),(1,2)],3)
=> 5
Description
The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1).
St000263: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 1 = 2 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> 4 = 5 - 1
([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
Description
The Szeged index of a graph.
St000265: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 1 = 2 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> 4 = 5 - 1
([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
Description
The Wiener index of a graph. This is the sum of the distances of all pairs of vertices.
Matching statistic: St000550
Mp00266: Graphs connected vertex partitionsLattices
St000550: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 5
Description
The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
Matching statistic: St000551
Mp00266: Graphs connected vertex partitionsLattices
St000551: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 5
Description
The number of left modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
Mp00203: Graphs coneGraphs
St001592: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([(0,1)],2)
=> 1
([],2)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
Description
The maximal number of simple paths between any two different vertices of a graph.
Mp00156: Graphs line graphGraphs
St001706: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],0)
=> 1
([],2)
=> ([],0)
=> 1
([(0,1)],2)
=> ([],1)
=> 2
([],3)
=> ([],0)
=> 1
([(1,2)],3)
=> ([],1)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 4
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 5
Description
The number of closed sets in a graph. A subset $S$ of the set of vertices is a closed set, if for any pair of distinct elements of $S$ the intersection of the corresponding neighbourhoods is a subset of $S$: $$ \forall a, b\in S: N(a)\cap N(b) \subseteq S. $$
Matching statistic: St001619
Mp00266: Graphs connected vertex partitionsLattices
St001619: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 2 = 1 + 1
([],2)
=> ([],1)
=> 2 = 1 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> 3 = 2 + 1
([],3)
=> ([],1)
=> 2 = 1 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 5 = 4 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 6 = 5 + 1
Description
The number of non-isomorphic sublattices of a lattice.
Matching statistic: St001666
Mp00266: Graphs connected vertex partitionsLattices
St001666: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 2 = 1 + 1
([],2)
=> ([],1)
=> 2 = 1 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> 3 = 2 + 1
([],3)
=> ([],1)
=> 2 = 1 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 5 = 4 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 6 = 5 + 1
Description
The number of non-isomorphic subposets of a lattice which are lattices.
The following 241 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000189The number of elements in the poset. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001717The largest size of an interval in a poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000009The charge of a standard tableau. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St001127The sum of the squares of the parts of a partition. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001764The number of non-convex subsets of vertices in a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000169The cocharge of a standard tableau. St000300The number of independent sets of vertices of a graph. St000330The (standard) major index of a standard tableau. St000420The number of Dyck paths that are weakly above a Dyck path. St001697The shifted natural comajor index of a standard Young tableau. St000418The number of Dyck paths that are weakly below a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St001645The pebbling number of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000086The number of subgraphs. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000468The Hosoya index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000454The largest eigenvalue of a graph if it is integral. St000631The number of distinct palindromic decompositions of a binary word. St000656The number of cuts of a poset. St000741The Colin de Verdière graph invariant. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000867The sum of the hook lengths in the first row of an integer partition. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001281The normalized isoperimetric number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000422The energy of a graph, if it is integral. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000012The area of a Dyck path. St000026The position of the first return of a Dyck path. St000038The product of the heights of the descending steps of a Dyck path. St000079The number of alternating sign matrices for a given Dyck path. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000391The sum of the positions of the ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000438The position of the last up step in a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000792The Grundy value for the game of ruler on a binary word. St000981The length of the longest zigzag subpath. St000984The number of boxes below precisely one peak. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001523The degree of symmetry of a Dyck path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001809The index of the step at the first peak of maximal height in a Dyck path. St001930The weak major index of a binary word. St001956The comajor index for set-valued two-row standard Young tableaux. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000460The hook length of the last cell along the main diagonal of an integer partition. St000699The toughness times the least common multiple of 1,. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St001626The number of maximal proper sublattices of a lattice. St000010The length of the partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000048The multinomial of the parts of a partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000117The number of centered tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000228The size of a partition. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000548The number of different non-empty partial sums of an integer partition. St000627The exponent of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000770The major index of an integer partition when read from bottom to top. St000784The maximum of the length and the largest part of the integer partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000876The number of factors in the Catalan decomposition of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000935The number of ordered refinements of an integer partition. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000992The alternating sum of the parts of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001267The length of the Lyndon factorization of the binary word. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001361The number of lattice paths of the same length that stay weakly above a Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001485The modular major index of a binary word. St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001527The cyclic permutation representation number of an integer partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001732The number of peaks visible from the left. St001808The box weight or horizontal decoration of a Dyck path. St001814The number of partitions interlacing the given partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001884The number of borders of a binary word. St001885The number of binary words with the same proper border set. St001910The height of the middle non-run of a Dyck path. St001959The product of the heights of the peaks of a Dyck path.