Your data matches 20 different statistics following compositions of up to 3 maps.
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Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
St000785: Graphs ⟶ 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)
=> ([(1,2)],3)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 3
['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)
=> 2
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: St000549
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000549: 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]
=> 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 3
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [5,1]
=> 2
Description
The number of odd partial sums of an integer partition.
Matching statistic: St000149
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00313: Integer partitions Glaisher-Franklin inverseInteger partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [3,1]
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [5,1]
=> 2 = 3 - 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,1,1,1]
=> 1 = 2 - 1
Description
The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1].
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,2,1,1]
=> 2 = 3 - 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,3]
=> 1 = 2 - 1
Description
The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000257
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,2,1,1]
=> 2 = 3 - 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,3]
=> 1 = 2 - 1
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St000481
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,2,1,1]
=> 2 = 3 - 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,3]
=> 1 = 2 - 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000506
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000506: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 3 - 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [5,1]
=> 1 = 2 - 1
Description
The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St000636
Mp00148: Finite Cartan types to root posetPosets
Mp00074: Posets to graphGraphs
Mp00203: Graphs coneGraphs
St000636: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 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)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
['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)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
Description
The hull number of a graph. The convex hull of a set of vertices $S$ of a graph is the smallest set $h(S)$ such that for any pair $u,v\in h(S)$ all vertices on a shortest path from $u$ to $v$ are also in $h(S)$. The hull number is the size of the smallest set $S$ such that $h(S)$ is the set of all vertices.
Matching statistic: St000671
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00111: Graphs complementGraphs
St000671: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 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)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
Description
The maximin edge-connectivity for choosing a subgraph. This is $\max_X \min(\lambda(G[X]), \lambda(G[V\setminus X]))$, where $X$ ranges over all subsets of the vertex set $V$ and $\lambda$ is the edge-connectivity of a graph.
Matching statistic: St001091
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,2,1,1]
=> 2 = 3 - 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,3]
=> 1 = 2 - 1
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001638The book thickness of a graph. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001964The interval resolution global dimension of a poset. St001624The breadth of a lattice. St001783The number of odd automorphisms of a graph. 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 $eA$. St001578The minimal number of edges to add or remove to make a graph a line graph.