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Your data matches 96 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 = 0 + 4
['A',2]
=> 8 = 4 + 4
['B',2]
=> 8 = 4 + 4
['G',2]
=> 8 = 4 + 4
['A',3]
=> 8 = 4 + 4
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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001248: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> [1,1]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 4
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 4
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> 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
Description
Sum of the even parts of a partition.
Matching statistic: St001109
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 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 = 4 + 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 = 4 + 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 = 4 + 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 = 4 + 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: St000633
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Mp00148: Finite Cartan types —to root poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
St000633: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Values
['A',1]
=> ([],1)
=> ? = 0 - 2
['A',2]
=> ([(0,2),(1,2)],3)
=> 2 = 4 - 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 4 - 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 4 - 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 2 = 4 - 2
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
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Mp00148: Finite Cartan types —to root poset⟶ Posets
St000642: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
St000642: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Values
['A',1]
=> ([],1)
=> ? = 0 - 2
['A',2]
=> ([(0,2),(1,2)],3)
=> 2 = 4 - 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 4 - 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 4 - 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 2 = 4 - 2
Description
The size of the smallest orbit of antichains under Panyushev complementation.
Matching statistic: St000914
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Mp00148: Finite Cartan types —to root poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
St000914: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Values
['A',1]
=> ([],1)
=> ? = 0 - 3
['A',2]
=> ([(0,2),(1,2)],3)
=> 1 = 4 - 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 4 - 3
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 1 = 4 - 3
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 4 - 3
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: St001630
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(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> ? = 0 - 2
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 4 - 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 4 - 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 4 - 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> ? = 0 - 2
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 4 - 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 4 - 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 4 - 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000706
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Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Values
['A',1]
=> ([],1)
=> [1]
=> ? = 0 - 3
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1 = 4 - 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1 = 4 - 3
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 1 = 4 - 3
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 1 = 4 - 3
Description
The product of the factorials of the multiplicities of an integer partition.
Matching statistic: St000993
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Values
['A',1]
=> ([],1)
=> [1]
=> ? = 0 - 3
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1 = 4 - 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1 = 4 - 3
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 1 = 4 - 3
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 1 = 4 - 3
Description
The multiplicity of the largest part of an integer partition.
The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001568The smallest positive integer that does not appear twice in the partition. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000699The toughness times the least common multiple of 1,. St000929The constant term of the character polynomial of an integer partition. St001281The normalized isoperimetric number 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. St001834The number of non-isomorphic minors of a graph. St000137The Grundy value of an integer partition. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000659The number of rises of length at least 2 of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001498The normalised height of a Nakayama algebra with magnitude 1. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001890The maximum magnitude of the Möbius function of a poset. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000145The Dyson rank of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000455The second largest eigenvalue of a graph if it is integral. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000567The sum of the products of all pairs of parts. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001095The number of non-isomorphic posets with precisely one further covering relation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001703The villainy of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
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