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Your data matches 39 different statistics following compositions of up to 3 maps.
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Matching statistic: St001891
St001891: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 4
['A',2]
=> 8
['B',2]
=> 8
['G',2]
=> 8
['A',3]
=> 8
['B',3]
=> 10
['C',3]
=> 10
Description
The size of a smallest Eulerian poset which does not appear as an interval in the Bruhat order of the Weyl group.
A bounded and graded poset is Eulerian if every non-trivial interval has the same number of elements of even and odd rank.
It is known that every interval of a Bruhat order is Eulerian. This statistic yields the minimal cardinality of an Eulerian poset not appearing in the Bruhat order.
Matching statistic: St001914
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> [1]
=> []
=> ? = 4 - 7
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 8 - 7
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 1 = 8 - 7
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> 1 = 8 - 7
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 8 - 7
['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]
=> [3,1]
=> 3 = 10 - 7
['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]
=> [3,1]
=> 3 = 10 - 7
Description
The size of the orbit of an integer partition in Bulgarian solitaire.
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.
This statistic returns the number of partitions that can be obtained from the given partition.
Matching statistic: St001392
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> [1]
=> []
=> ? = 4 - 8
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> 0 = 8 - 8
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 0 = 8 - 8
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> 0 = 8 - 8
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 0 = 8 - 8
['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]
=> [3,1]
=> 2 = 10 - 8
['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]
=> [3,1]
=> 2 = 10 - 8
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001109
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 4 - 2
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6 = 8 - 2
['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)
=> 6 = 8 - 2
['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)
=> 6 = 8 - 2
['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)
=> 6 = 8 - 2
['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),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {10,10} - 2
['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),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {10,10} - 2
Description
The number of proper colourings of a graph with as few colours as possible.
By definition, this is the evaluation of the chromatic polynomial at the first nonnegative integer which is not a zero of the polynomial.
Matching statistic: St001248
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001248: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001248: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> [2]
=> [1,1]
=> 0 = 4 - 4
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 4 = 8 - 4
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 4 = 8 - 4
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> 4 = 8 - 4
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 4 = 8 - 4
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> [6,6,6,2]
=> [4,4,3,3,3,3]
=> ? ∊ {10,10} - 4
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> [6,6,6,2]
=> [4,4,3,3,3,3]
=> ? ∊ {10,10} - 4
Description
Sum of the even parts of a partition.
Matching statistic: St000633
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 57%●distinct values known / distinct values provided: 33%
St000633: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 57%●distinct values known / distinct values provided: 33%
Values
['A',1]
=> ([],1)
=> ? = 4 - 6
['A',2]
=> ([(0,2),(1,2)],3)
=> 2 = 8 - 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 8 - 6
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 8 - 6
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 2 = 8 - 6
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {10,10} - 6
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {10,10} - 6
Description
The size of the automorphism group of a poset.
A poset automorphism is a permutation of the elements of the poset preserving the order relation.
Matching statistic: St000642
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
St000642: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 57%●distinct values known / distinct values provided: 33%
St000642: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 57%●distinct values known / distinct values provided: 33%
Values
['A',1]
=> ([],1)
=> ? = 4 - 6
['A',2]
=> ([(0,2),(1,2)],3)
=> 2 = 8 - 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 8 - 6
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 8 - 6
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 2 = 8 - 6
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {10,10} - 6
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {10,10} - 6
Description
The size of the smallest orbit of antichains under Panyushev complementation.
Matching statistic: St000910
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
St000910: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
St000910: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> ? = 4 - 6
['A',2]
=> ([(0,2),(1,2)],3)
=> 2 = 8 - 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 8 - 6
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 8 - 6
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 10 - 6
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {8,10} - 6
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {8,10} - 6
Description
The number of maximal chains of minimal length in a poset.
Matching statistic: St000914
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 57%●distinct values known / distinct values provided: 33%
St000914: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 57%●distinct values known / distinct values provided: 33%
Values
['A',1]
=> ([],1)
=> ? = 4 - 7
['A',2]
=> ([(0,2),(1,2)],3)
=> 1 = 8 - 7
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 8 - 7
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 1 = 8 - 7
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 8 - 7
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {10,10} - 7
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {10,10} - 7
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\
-\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\
0&\text{otherwise}.
\end{cases}
$$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$
\sum_{x\leq y} \mu(x,y).
$$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Matching statistic: St001095
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
St001095: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
St001095: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> ? = 4 - 8
['A',2]
=> ([(0,2),(1,2)],3)
=> 0 = 8 - 8
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 0 = 8 - 8
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 0 = 8 - 8
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 2 = 10 - 8
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {8,10} - 8
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ? ∊ {8,10} - 8
Description
The number of non-isomorphic posets with precisely one further covering relation.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001118The acyclic chromatic index of a graph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001834The number of non-isomorphic minors of a graph. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. 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 $eA$. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000456The monochromatic index of a connected graph. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001592The maximal number of simple paths between any two different vertices of a graph. St001890The maximum magnitude of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000508Eigenvalues of the random-to-random operator acting on a simple module. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001703The villainy of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000509The diagonal index (content) of a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
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