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Your data matches 156 different statistics following compositions of up to 3 maps.
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Matching statistic: St000718
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 2
([],3)
=> 0
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> 3
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St001279
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 0
([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> [2]
=> 2
([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> [2,1]
=> 2
([(0,2),(1,2)],3)
=> [3]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001458
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The rank of the adjacency matrix of a graph.
Matching statistic: St001459
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The number of zero columns in the nullspace of a graph.
Matching statistic: St001690
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 0
([],2)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(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)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path.
Put differently, for every vertex $v$ of such a path $P$, there is a vertex $w\in P$ and a vertex $u\not\in P$ such that $(v, u)$ and $(u, w)$ are edges.
The length of such a path is $0$ if the graph is a forest.
It is maximal, if and only if the graph is obtained from a graph $H$ with a Hamiltonian path by joining a new vertex to each of the vertices of $H$.
Matching statistic: St001796
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1).
Matching statistic: St000171
Values
([],1)
=> ([(0,1)],2)
=> ([],1)
=> 0
([],2)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000380
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> []
=> 0
([],2)
=> [1,1]
=> [1]
=> 2
([(0,1)],2)
=> [2]
=> []
=> 0
([],3)
=> [1,1,1]
=> [1,1]
=> 3
([(1,2)],3)
=> [2,1]
=> [1]
=> 2
([(0,2),(1,2)],3)
=> [2,2]
=> [2]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> 0
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St000987
Values
([],1)
=> ([(0,1)],2)
=> ([],1)
=> 0
([],2)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001119
Values
([],1)
=> ([(0,1)],2)
=> ([],1)
=> 0
([],2)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The length of a shortest maximal path in a graph.
The following 146 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001120The length of a longest path in a graph. St001214The aft of an integer partition. St001391The disjunction number of a graph. St001110The 3-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001345The Hamming dimension of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001706The number of closed sets in a graph. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000463The number of admissible inversions of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000825The sum of the major and the inverse major index of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000921The number of internal inversions of a binary word. St000979Half of MacMahon's equal index of a Dyck path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001379The number of inversions plus the major index of a permutation. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000240The number of indices that are not small excedances. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000763The sum of the positions of the strong records of an integer composition. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001468The smallest fixpoint of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000673The number of non-fixed points of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000652The maximal difference between successive positions of a permutation. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000477The weight of a partition according to Alladi. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000741The Colin de Verdière graph invariant. St000815The number of semistandard Young tableaux of partition weight of given shape. St000874The position of the last double rise in a Dyck path. St000906The length of the shortest maximal chain in a poset. St000976The sum of the positions of double up-steps of a Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000422The energy of a graph, if it is integral. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000012The area of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000144The pyramid weight of the Dyck path. St000347The inversion sum of a binary word. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000420The number of Dyck paths that are weakly above a Dyck path. St000439The position of the first down step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000969We 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 $c_{n-1}$. St000981The length of the longest zigzag subpath. St000983The length of the longest alternating subword. St000984The number of boxes below precisely one peak. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001437The flex of a binary word. St001480The number of simple summands of the module J^2/J^3. St001500The global 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001721The degree of a binary word. St001808The box weight or horizontal decoration of a Dyck path. St001930The weak major index of a binary word. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another 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. St001754The number of tolerances of a finite lattice. St000014The number of parking functions supported by a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000294The number of distinct factors of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000995The largest even part of an integer partition. St001018Sum of projective dimension of the indecomposable injective modules of 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. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001248Sum of the even parts of a partition. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001400The total number of Littlewood-Richardson tableaux of given shape. St001498The normalised height of a Nakayama algebra with magnitude 1. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001669The number of single rises in a Dyck path. St001814The number of partitions interlacing the given partition. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001956The comajor index for set-valued two-row standard Young tableaux.
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