Your data matches 216 different statistics following compositions of up to 3 maps.
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Matching statistic: St000627
(load all 49 compositions to match this statistic)
Mapping
St000627: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 = 2 - 1
1 => 1 = 2 - 1
00 => 2 = 3 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 3 = 4 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 1 = 2 - 1
111 => 3 = 4 - 1
Description
The exponent of a binary word. This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called '''frequency'''.
Matching statistic: St000907
(load all 3 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => 2
1 => ([(0,1)],2)
 => 2
00 => ([(0,2),(2,1)],3)
 => 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
11 => ([(0,2),(2,1)],3)
 => 3
000 => ([(0,3),(2,1),(3,2)],4)
 => 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => 4
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000657
(load all 11 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 2 = 3 - 1
01 => [1,1] => 1 = 2 - 1
10 => [1,1] => 1 = 2 - 1
11 => [2] => 2 = 3 - 1
000 => [3] => 3 = 4 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 1 = 2 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 1 = 2 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 3 = 4 - 1
Description
The smallest part of an integer composition.
Matching statistic: St000876
(load all 3 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 2 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 2 = 3 - 1
01 => 10 => 1 = 2 - 1
10 => 10 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 3 = 4 - 1
001 => 100 => 1 = 2 - 1
010 => 100 => 1 = 2 - 1
011 => 110 => 1 = 2 - 1
100 => 100 => 1 = 2 - 1
101 => 110 => 1 = 2 - 1
110 => 110 => 1 = 2 - 1
111 => 111 => 3 = 4 - 1
Description
The number of factors in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000899
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 1 = 2 - 1
01 => [1,1] => 2 = 3 - 1
10 => [1,1] => 2 = 3 - 1
11 => [2] => 1 = 2 - 1
000 => [3] => 1 = 2 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 3 = 4 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 3 = 4 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 1 = 2 - 1
Description
The maximal number of repetitions of an integer composition. This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000900
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000900: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 1 = 2 - 1
01 => [1,1] => 2 = 3 - 1
10 => [1,1] => 2 = 3 - 1
11 => [2] => 1 = 2 - 1
000 => [3] => 1 = 2 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 3 = 4 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 3 = 4 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 1 = 2 - 1
Description
The minimal number of repetitions of a part in an integer composition. This is the smallest letter in the word obtained by applying the delta morphism.
Matching statistic: St000902
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000902: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 1 = 2 - 1
01 => [1,1] => 2 = 3 - 1
10 => [1,1] => 2 = 3 - 1
11 => [2] => 1 = 2 - 1
000 => [3] => 1 = 2 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 3 = 4 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 3 = 4 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 1 = 2 - 1
Description
The minimal number of repetitions of an integer composition.
Matching statistic: St000904
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 1 = 2 - 1
01 => [1,1] => 2 = 3 - 1
10 => [1,1] => 2 = 3 - 1
11 => [2] => 1 = 2 - 1
000 => [3] => 1 = 2 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 3 = 4 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 3 = 4 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 1 = 2 - 1
Description
The maximal number of repetitions of an integer composition.
Matching statistic: St001236
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 1 = 2 - 1
01 => [1,1] => 2 = 3 - 1
10 => [1,1] => 2 = 3 - 1
11 => [2] => 1 = 2 - 1
000 => [3] => 1 = 2 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 3 = 4 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 3 = 4 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 1 = 2 - 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Matching statistic: St001267
(load all 5 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 2 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 2 = 3 - 1
01 => 01 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 3 = 4 - 1
001 => 001 => 1 = 2 - 1
010 => 001 => 1 = 2 - 1
011 => 011 => 1 = 2 - 1
100 => 001 => 1 = 2 - 1
101 => 011 => 1 = 2 - 1
110 => 011 => 1 = 2 - 1
111 => 111 => 3 = 4 - 1
Description
The length of the Lyndon factorization of the binary word. The Lyndon factorization of a finite word w is its unique factorization as a non-increasing product of Lyndon words, i.e., $w = l_1\dots l_n$ where each $l_i$ is a Lyndon word and $l_1 \geq\dots\geq l_n$.
The following 206 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001437The flex of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000850The number of 1/2-balanced pairs in a poset. St000315The number of isolated vertices of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001691The number of kings in a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000160The multiplicity of the smallest part of a partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000543The size of the conjugacy class of a binary word. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000775The multiplicity of the largest eigenvalue in a graph. St000808The number of up steps of the associated bargraph. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective 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. 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. 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 $g$. St001481The minimal height of a peak of a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000552The number of cut vertices of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001091The number of parts in an integer partition whose next smaller part has the same size. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000148The number of odd parts of a partition. St000439The position of the first down step of a Dyck path. St000475The number of parts equal to 1 in a partition. St000906The length of the shortest maximal chain in a poset. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. 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. 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. St001654The monophonic hull number of a graph. St001814The number of partitions interlacing the given partition. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000056The decomposition (or block) number of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000273The domination number of a graph. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. 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. 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. St000482The (zero)-forcing number of a graph. St000529The number of permutations whose descent word is the given binary word. St000544The cop number of a graph. St000626The minimal period of a binary word. St000700The protection number of an ordered tree. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000883The number of longest increasing subsequences of a permutation. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000916The packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000935The number of ordered refinements of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in 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. 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. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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. 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. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001286The annihilation number of a graph. 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. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001372The length of a longest cyclic run of ones of a binary word. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. 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. St001463The number of distinct columns in the nullspace of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001527The cyclic permutation representation number of an integer partition. St001614The cyclic permutation representation number of a skew partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001675The number of parts equal to the part in the reversed composition. St001739The number of graphs with the same edge polytope as the given graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000024The number of double up and double down steps of a Dyck path. St000210Minimum over maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000376The bounce deficit of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St000877The depth of the binary word interpreted as a path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001130The number of two successive successions in a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001479The number of bridges of a graph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001689The number of celebrities in a graph. St001777The number of weak descents in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001826The maximal number of leaves on a vertex of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000993The multiplicity of the largest part of an integer partition. St000478Another weight of a partition according to Alladi. St000487The length of the shortest cycle of a permutation. St000729The minimal arc length of a set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000937The number of positive values of the symmetric group character corresponding to the partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001061The number of indices that are both descents and recoils of a permutation. St001060The distinguishing index of a graph. St000741The Colin de Verdière graph invariant. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001330The hat guessing number of a 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001624The breadth of a lattice. St001644The dimension of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000454The largest eigenvalue of a graph if it is integral. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral.