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Your data matches 280 different statistics following compositions of up to 3 maps.
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Matching statistic: St000812
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000812: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000812: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> 1
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
([(4,5)],6)
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 3
([(5,6)],7)
=> [1]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> 3
([(3,6),(4,5)],7)
=> [1,1]
=> 3
([(4,5),(4,6),(5,6)],7)
=> [3]
=> 3
Description
The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions.
For example, $h_{11} = 2m_{11} + m_2$, so the statistic on the partition $11$ is 3.
Matching statistic: St000422
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(4,5)],6)
=> ([(4,5)],6)
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
([(5,6)],7)
=> ([(5,6)],7)
=> 2 = 1 + 1
([(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> 4 = 3 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000468
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(4,5)],6)
=> ([(4,5)],6)
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
([(5,6)],7)
=> ([(5,6)],7)
=> 2 = 1 + 1
([(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> 4 = 3 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
Description
The Hosoya index of a graph.
This is the total number of matchings in the graph.
Matching statistic: St001618
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(2,3)],4)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(3,4)],5)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(4,5)],6)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(5,6)],7)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
Description
The cardinality of the Frattini sublattice of a lattice.
The Frattini sublattice is the intersection of all proper maximal sublattices of the lattice.
Matching statistic: St000016
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000016: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000016: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [2]
=> [[1,2]]
=> 1
([(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> 1
([(0,2),(1,2)],3)
=> [2,2]
=> [[1,2],[3,4]]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 3
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 3
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 3
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 1
([(4,6),(5,6)],7)
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> 3
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 3
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 3
Description
The number of attacking pairs of a standard tableau.
Note that this is actually a statistic on the underlying partition.
A pair of cells $(c, d)$ of a Young diagram (in English notation) is said to be attacking if one of the following conditions holds:
1. $c$ and $d$ lie in the same row with $c$ strictly to the west of $d$.
2. $c$ is in the row immediately to the south of $d$, and $c$ lies strictly east of $d$.
Matching statistic: St000526
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
Description
The number of posets with combinatorially isomorphic order polytopes.
Matching statistic: St000867
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000867: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000867: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1]
=> 1
([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
([(2,3)],4)
=> [1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(3,4)],5)
=> [1]
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(4,5)],6)
=> [1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 3
([(5,6)],7)
=> [1]
=> [1]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> [2]
=> 3
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> 3
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> 3
Description
The sum of the hook lengths in the first row of an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition.
Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.
Matching statistic: St001228
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(2,3)],4)
=> [1]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(3,4)],5)
=> [1]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(4,5)],6)
=> [1]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(5,6)],7)
=> [1]
=> [1,0]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> 3
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> 3
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,0,1,0,1,0]
=> 3
Description
The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001254
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(2,3)],4)
=> [1]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(3,4)],5)
=> [1]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(4,5)],6)
=> [1]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(5,6)],7)
=> [1]
=> [1,0]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> 3
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> 3
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,0,1,0,1,0]
=> 3
Description
The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St001694
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001694: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001694: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 3
([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(4,5)],6)
=> [1,5] => ([(4,5)],6)
=> 1
([(3,5),(4,5)],6)
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
([(2,5),(3,4)],6)
=> [2,4] => ([(3,5),(4,5)],6)
=> 3
([(3,4),(3,5),(4,5)],6)
=> [2,4] => ([(3,5),(4,5)],6)
=> 3
([(5,6)],7)
=> [1,6] => ([(5,6)],7)
=> 1
([(4,6),(5,6)],7)
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 3
([(3,6),(4,5)],7)
=> [2,5] => ([(4,6),(5,6)],7)
=> 3
([(4,5),(4,6),(5,6)],7)
=> [2,5] => ([(4,6),(5,6)],7)
=> 3
Description
The number of maximal dissociation sets in a graph.
The following 270 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000511The number of invariant subsets when acting with a permutation of given cycle type. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St000146The Andrews-Garvan crank of a partition. St000026The position of the first return of a Dyck path. St000347The inversion sum of a binary word. St000391The sum of the positions of the ones in a binary word. St000420The number of Dyck paths that are weakly above a Dyck path. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000626The minimal period of a binary word. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000869The sum of the hook lengths of an integer partition. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000976The sum of the positions of double up-steps of a Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001721The degree of a binary word. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001930The weak major index of a binary word. St000120The number of left tunnels of a Dyck path. St000293The number of inversions of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000395The sum of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000979Half of MacMahon's equal index of a Dyck path. St000995The largest even part of an integer partition. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001161The major index north count of a Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000438The position of the last up step in a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St000037The sign of a permutation. St001869The maximum cut size of a graph. St000915The Ore degree of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000939The number of characters of the symmetric group whose value on the partition is positive. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000675The number of centered multitunnels of a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001568The smallest positive integer that does not appear twice in the partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000444The length of the maximal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000981The length of the longest zigzag subpath. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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 $eA$. 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 $eA$. 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 $eA$. St000442The maximal area to the right of an up step of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. 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. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St000467The hyper-Wiener index of a connected graph. St000699The toughness times the least common multiple of 1,. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001281The normalized isoperimetric number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001875The number of simple modules with projective dimension at most 1. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. 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. St001877Number of indecomposable injective modules with projective dimension 2. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. 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. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001651The Frankl number of a lattice. St001118The acyclic chromatic index of a graph. 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. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St000741The Colin de Verdière graph invariant. St001545The second Elser number of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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