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Your data matches 98 different statistics following compositions of up to 3 maps.
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Matching statistic: St001369
St001369: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 1 = 0 + 1
['A',2]
=> 1 = 0 + 1
['B',2]
=> 2 = 1 + 1
['G',2]
=> 3 = 2 + 1
Description
The largest coefficient in the highest root in the root system of a Cartan type.
Matching statistic: St000149
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> 0
['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]
=> 2
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000313
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
Description
The number of degree 2 vertices of a graph.
A vertex has degree 2 if and only if it lies on a unique maximal path.
Matching statistic: St000697
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000697: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000697: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 0
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 2
Description
The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $3$-rim hooks that are removed in this process to obtain a $3$-core.
Matching statistic: St001056
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(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
Description
The Grundy value for the game of deleting vertices of a graph until it has no edges.
Matching statistic: St001646
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['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)
=> 2
Description
The number of edges that can be added without increasing the maximal degree of a graph.
This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on $n$ vertices, which has maximal degree $n-1$ and therefore has statistic $\binom{n-1}{2}$.
Matching statistic: St001912
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001912: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001912: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 2
Description
The length of the preperiod in Bulgarian solitaire corresponding to an integer partition.
Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
Matching statistic: St000785
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 0 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 3 = 2 + 1
Description
The number of distinct colouring schemes of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the number of distinct partitions that occur.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$.
Matching statistic: St001442
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001442: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001442: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 1 = 0 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 3 = 2 + 1
Description
The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.
Matching statistic: St001587
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 1 = 0 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 3 = 2 + 1
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
The following 88 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000671The maximin edge-connectivity for choosing a subgraph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001214The aft of an integer partition. St001252Half the sum of the even parts of a partition. St001323The independence gap of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001657The number of twos in an integer partition. St001689The number of celebrities in a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000482The (zero)-forcing number of a graph. St000549The number of odd partial sums of an integer partition. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001111The weak 2-dynamic chromatic number of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001527The cyclic permutation representation number of an integer partition. St001739The number of graphs with the same edge polytope as the given graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001949The rigidity index of a graph. St000636The hull number of a graph. St001315The dissociation number of a graph. St000478Another weight of a partition according to Alladi. St001964The interval resolution global dimension of a poset. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000936The number of even 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000454The largest eigenvalue of a graph if it is integral. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000928The sum of the coefficients of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001578The minimal number of edges to add or remove to make a graph a line graph. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001644The dimension of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000741The Colin de Verdière graph invariant. St001330The hat guessing number of a graph. St001386The number of prime labellings of a graph. St001391The disjunction number of a graph. 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). St001568The smallest positive integer that does not appear twice in the partition. St001624The breadth of a lattice. St001642The Prague dimension of a graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001783The number of odd automorphisms of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000299The number of nonisomorphic vertex-induced subtrees. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
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