Your data matches 86 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000855
Mapping
St000855: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => 1 = 2 - 1
['A',2]
 => 1 = 2 - 1
['B',2]
 => 2 = 3 - 1
['G',2]
 => 4 = 5 - 1
['A',3]
 => 1 = 2 - 1
['B',3]
 => 3 = 4 - 1
['C',3]
 => 3 = 4 - 1
Description
The number of full-support reflections in the Weyl group of a finite Cartan type. A reflection has full support if any (or all) reduced words for it in simple reflections use all simple reflections. This number is given by $\frac{nh}{|W|}d_1^*\cdots d_{n-1}^*$ where $n$ is the rank, $h$ is the Coxeter number, $W$ is the Weyl group, and $d_1^* \geq \ldots \geq d_{n-1}^* \geq d_n^* = 0$ are the codegrees of the Weyl group of a Cartan type.
Matching statistic: St000145
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => 0 = 2 - 2
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 0 = 2 - 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 1 = 3 - 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 3 = 5 - 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 0 = 2 - 2
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 2 = 4 - 2
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 2 = 4 - 2
Description
The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000439
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 2
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 3
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 5
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 2
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 4
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 4
Description
The position of the first down step of a Dyck path.
Matching statistic: St001504
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 2
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 3
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 5
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 2
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 4
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 4
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000013
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 2 - 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4 = 5 - 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000025
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 2 - 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4 = 5 - 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000443
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 2 - 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4 = 5 - 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
Description
The number of long tunnels of a Dyck path. A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St000444
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 2 - 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4 = 5 - 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000876
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => 10 => 1 = 2 - 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1010 => 1 = 2 - 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 10010 => 2 = 3 - 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 1000010 => 4 = 5 - 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 101010 => 1 = 2 - 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 10010010 => 3 = 4 - 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 10010010 => 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: St001007
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 2 - 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4 = 5 - 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
The following 76 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001498The normalised height of a Nakayama algebra with magnitude 1. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001584The area statistic between a Dyck path and its bounce path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000907The number of maximal antichains of minimal length in a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000315The number of isolated vertices of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001691The number of kings in a graph. St001623The number of doubly irreducible elements of a lattice. St001672The restrained domination 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. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001479The number of bridges of a graph. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001820The size of the image of the pop stack sorting operator. St001826The maximal number of leaves on a vertex of a graph. St001949The rigidity index of a graph. St000313The number of degree 2 vertices of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001347The number of pairs of vertices of a graph having the same neighbourhood. St000526The number of posets with combinatorially isomorphic order polytopes. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001875The number of simple modules with projective dimension at most 1. St000693The modular (standard) major index of a standard tableau. St000906The length of the shortest maximal chain in a poset. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000643The size of the largest orbit of antichains under Panyushev complementation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001881The number of factors of a lattice as a Cartesian product of lattices. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001964The interval resolution global dimension of a poset. St001060The distinguishing index of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001624The breadth of a lattice. St001754The number of tolerances of a finite lattice. St001783The number of odd automorphisms of a graph. St000509The diagonal index (content) of a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph.