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Your data matches 66 different statistics following compositions of up to 3 maps.
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Matching statistic: St000860
St000860: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 2 = 1 + 1
['A',2]
=> 1 = 0 + 1
['B',2]
=> 2 = 1 + 1
['G',2]
=> 2 = 1 + 1
['A',3]
=> 1 = 0 + 1
['B',3]
=> 2 = 1 + 1
['C',3]
=> 2 = 1 + 1
Description
The size of the center of the Weyl group of a finite Cartan type.
Matching statistic: St000811
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000811: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000811: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 0
['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
['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
Description
The 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.
For example, p22=s1111−s211+2s22−s31+s4, so the statistic on the partition 22 is 2.
This is also the sum of the character values at the given conjugacy class over all irreducible characters of the symmetric group. [2]
For a permutation π of given cycle type, this is also the number of permutations whose square equals π. [2]
Matching statistic: St001484
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,2,1,1]
=> 0
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,3]
=> 0
['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]
=> [3,2,2,1,1]
=> 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]
=> [3,2,2,1,1]
=> 1
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St001939
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [3,1]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [5,1]
=> 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,1,1,1]
=> 0
['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]
=> [5,3,1]
=> 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]
=> [5,3,1]
=> 1
Description
The number of parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001940
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [3,1]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [5,1]
=> 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,1,1,1]
=> 0
['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]
=> [5,3,1]
=> 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]
=> [5,3,1]
=> 1
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001568
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> [1]
=> ? = 1 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2 = 1 + 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> [7,1,1]
=> 2 = 1 + 1
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> [7,1,1]
=> 2 = 1 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001057
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {0,1}
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {0,1}
Description
The Grundy value of the game of creating an independent set in a graph.
Two players alternately add a vertex to an initially empty set, which is not adjacent to any of the vertices it already contains.
Alternatively, the game can be described as starting with a graph, the players remove vertices together with their neighbors, until the graph is empty.
Matching statistic: St000315
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 0
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1}
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1}
Description
The number of isolated vertices of a graph.
Matching statistic: St000986
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 0
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,1}
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,1}
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001070
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {1,1}
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {1,1}
Description
The absolute value of the derivative of the chromatic polynomial of the graph at 1.
This is closely related to Crapo's beta invariant, the only difference being the value for the graphs without edges.
The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001341The number of edges in the center of a graph. St001363The Euler characteristic of a graph according to Knill. St001691The number of kings in a graph. St000351The determinant of the adjacency matrix of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001118The acyclic chromatic index of a graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001820The size of the image of the pop stack sorting operator. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St000181The number of connected components of the Hasse diagram for the poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000286The number of connected components of the complement of a graph. St000311The number of vertices of odd degree in a graph. St000322The skewness of a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001577The minimal number of edges to add or remove to make a graph a cograph. St001871The number of triconnected components of a graph. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.
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