Your data matches 119 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000506
(load all 2 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => ([(0,2),(1,2)],3)
 => [2]
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => [2]
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [2]
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [2]
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [1,1]
 => 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [2]
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [2]
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [2]
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [2]
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [1,1]
 => 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [2]
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [2]
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 0
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [1,1]
 => 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [1,1]
 => 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [2]
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 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: St000621
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => ([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 0
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => 1
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].
Matching statistic: St000260
(load all 6 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 58%●distinct values known / distinct values provided: 12%
Values
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000379
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 57%●distinct values known / distinct values provided: 12%
Values
[2,3,1] => [1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[3,1,2] => [2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => [1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ? = 1
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[4,3,2,1] => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? = 1
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[1,4,5,3,2] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[1,5,3,2,4] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,5,4,2,3] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? = 1
[2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? = 1
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 1
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 1
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 1
[2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 1
[2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[2,3,5,4,1] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[2,4,3,1,5] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[2,4,3,5,1] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[2,4,5,3,1] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[2,5,3,4,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 0
[2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 1
[3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 1
[3,2,4,1,5] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[1,2,3,5,6,4] => [4,6,5,3,2,1] => ([(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)
 => ([],1)
 => ? = 0
[1,2,3,6,4,5] => [5,4,6,3,2,1] => ([(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)
 => ([],1)
 => ? = 0
[1,2,3,6,5,4] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 1
[1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(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)
 => ([],1)
 => ? = 0
[1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(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)
 => ([],1)
 => ? = 0
[1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,4,6,5,3] => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,5,3,4,6] => [6,4,3,5,2,1] => ([(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)
 => ([],1)
 => ? = 0
[1,2,5,3,6,4] => [4,6,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,5,4,3,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,5,4,6,3] => [3,6,4,5,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,5,6,4,3] => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(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)
 => ([],1)
 => ? = 0
[1,2,6,3,5,4] => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,6,4,3,5] => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,6,4,5,3] => [3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,6,5,3,4] => [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,2,6,5,4,3] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 0
[1,3,2,4,6,5] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 1
[1,3,2,5,4,6] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? = 1
[1,3,2,5,6,4] => [4,6,5,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ? = 1
[1,3,2,6,4,5] => [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ? = 1
[1,3,2,6,5,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ? = 1
Description
The number of Hamiltonian cycles in a graph. A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$. Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000456
(load all 6 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 41%●distinct values known / distinct values provided: 12%
Values
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ? = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ? = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ? = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 0 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 1 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 1 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 0 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 0 + 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 0 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 0 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 0 + 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 + 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 0 + 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
Mapping
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 27%●distinct values known / distinct values provided: 12%
Values
[2,3,1] => [1,2,3] => [3] => ([],3)
 => ? = 0
[3,1,2] => [3,1,2] => [3] => ([],3)
 => ? = 0
[3,2,1] => [2,1,3] => [3] => ([],3)
 => ? = 0
[1,3,4,2] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,4,2,3] => [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,4,3,2] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,1,4,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,3,1,4] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
 => ? = 0
[2,4,1,3] => [1,4,2,3] => [1,3] => ([(2,3)],4)
 => 0
[2,4,3,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
 => 0
[3,1,2,4] => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,2,1,4] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [2,1,3,4] => [4] => ([],4)
 => ? = 0
[3,4,1,2] => [4,1,2,3] => [4] => ([],4)
 => ? = 0
[3,4,2,1] => [3,1,2,4] => [4] => ([],4)
 => ? = 0
[4,1,2,3] => [3,4,1,2] => [4] => ([],4)
 => ? = 0
[4,1,3,2] => [4,3,1,2] => [1,3] => ([(2,3)],4)
 => 0
[4,2,1,3] => [2,4,1,3] => [4] => ([],4)
 => ? = 0
[4,2,3,1] => [2,3,1,4] => [4] => ([],4)
 => ? = 0
[4,3,1,2] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[4,3,2,1] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[1,2,4,5,3] => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => [2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,2,5,4] => [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,3,4,2,5] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => [4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,5,4,2] => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,2,3,5] => [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => [3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[1,4,3,2,5] => [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,3,5,2] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,5,2,3] => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[1,4,5,3,2] => [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,2,3,4] => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[1,5,2,4,3] => [3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,3,2,4] => [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,3,4,2] => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,4,2,3] => [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,5,4,3,2] => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[2,1,3,5,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,1,4,3,5] => [1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,1,4,5,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,1,5,3,4] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,1,5,4,3] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,3,1,4,5] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,3,4,1,5] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
 => ? = 0
[2,3,5,1,4] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[2,3,5,4,1] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[2,4,1,5,3] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[2,4,3,1,5] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[2,4,3,5,1] => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
 => 0
[2,4,5,1,3] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
 => 0
[2,4,5,3,1] => [1,4,2,3,5] => [1,4] => ([(3,4)],5)
 => 0
[2,5,1,3,4] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
 => 0
[2,5,1,4,3] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[2,5,3,1,4] => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
 => 0
[2,5,3,4,1] => [1,3,4,2,5] => [1,4] => ([(3,4)],5)
 => 0
[2,5,4,1,3] => [1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[2,5,4,3,1] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[3,1,2,4,5] => [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => [3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[3,1,4,2,5] => [4,1,3,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,4,5,2] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[3,1,5,2,4] => [4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[3,1,5,4,2] => [5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[3,2,1,4,5] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[3,2,4,1,5] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,2,4,5,1] => [2,1,3,4,5] => [5] => ([],5)
 => ? = 0
[3,2,5,1,4] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[3,2,5,4,1] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[3,4,1,2,5] => [4,1,2,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,4,1,5,2] => [5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[3,4,2,1,5] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,4,2,5,1] => [3,1,2,4,5] => [5] => ([],5)
 => ? = 0
[3,4,5,1,2] => [5,1,2,3,4] => [5] => ([],5)
 => ? = 0
[3,4,5,2,1] => [4,1,2,3,5] => [5] => ([],5)
 => ? = 0
[3,5,1,2,4] => [4,1,5,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[3,5,1,4,2] => [5,1,4,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[3,5,2,1,4] => [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[3,5,2,4,1] => [3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[3,5,4,1,2] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,5,4,2,1] => [4,1,3,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[4,1,2,3,5] => [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[4,1,2,5,3] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[4,1,3,2,5] => [4,3,1,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[4,1,3,5,2] => [5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[4,1,5,2,3] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[4,1,5,3,2] => [5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[4,2,3,5,1] => [2,3,1,4,5] => [5] => ([],5)
 => ? = 0
[4,2,5,1,3] => [2,5,1,3,4] => [5] => ([],5)
 => ? = 0
[4,2,5,3,1] => [2,4,1,3,5] => [5] => ([],5)
 => ? = 0
[4,3,1,5,2] => [5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000259
(load all 2 compositions to match this statistic)
Mapping
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 21%●distinct values known / distinct values provided: 12%
Values
[2,3,1] => [2,3,1] => [3] => ([],3)
 => ? = 0 + 2
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
 => ? = 0 + 2
[3,2,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,3,4,2] => [3,4,1,2] => [4] => ([],4)
 => ? = 0 + 2
[1,4,2,3] => [1,4,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0 + 2
[1,4,3,2] => [4,3,1,2] => [1,3] => ([(2,3)],4)
 => ? = 0 + 2
[2,1,4,3] => [2,4,1,3] => [4] => ([],4)
 => ? = 1 + 2
[2,3,1,4] => [2,3,1,4] => [4] => ([],4)
 => ? = 0 + 2
[2,3,4,1] => [2,3,4,1] => [4] => ([],4)
 => ? = 0 + 2
[2,4,1,3] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[2,4,3,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0 + 2
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
 => ? = 0 + 2
[3,2,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[3,2,4,1] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[3,4,1,2] => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[3,4,2,1] => [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,1,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[4,1,3,2] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,2,1,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[4,2,3,1] => [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,3,1,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,3,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,2,4,5,3] => [4,5,1,2,3] => [5] => ([],5)
 => ? = 0 + 2
[1,2,5,3,4] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,3,2,5,4] => [3,5,1,2,4] => [5] => ([],5)
 => ? = 1 + 2
[1,3,4,2,5] => [3,4,1,2,5] => [5] => ([],5)
 => ? = 0 + 2
[1,3,4,5,2] => [3,4,5,1,2] => [5] => ([],5)
 => ? = 0 + 2
[1,3,5,2,4] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,3,5,4,2] => [5,3,4,1,2] => [1,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,4,2,5,3] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,4,3,2,5] => [4,3,1,2,5] => [1,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,4,3,5,2] => [4,3,5,1,2] => [1,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,4,5,2,3] => [4,1,5,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[1,4,5,3,2] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,5,2,3,4] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[1,5,2,4,3] => [5,1,4,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[1,5,3,2,4] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,5,3,4,2] => [3,5,4,1,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[1,5,4,2,3] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,5,4,3,2] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,1,3,5,4] => [2,5,1,3,4] => [5] => ([],5)
 => ? = 1 + 2
[2,1,4,3,5] => [2,4,1,3,5] => [5] => ([],5)
 => ? = 1 + 2
[2,1,4,5,3] => [2,4,5,1,3] => [5] => ([],5)
 => ? = 1 + 2
[2,1,5,3,4] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1 + 2
[2,1,5,4,3] => [5,2,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1 + 2
[2,3,1,4,5] => [2,3,1,4,5] => [5] => ([],5)
 => ? = 0 + 2
[2,3,1,5,4] => [2,3,5,1,4] => [5] => ([],5)
 => ? = 1 + 2
[2,3,4,1,5] => [2,3,4,1,5] => [5] => ([],5)
 => ? = 0 + 2
[2,3,4,5,1] => [2,3,4,5,1] => [5] => ([],5)
 => ? = 0 + 2
[2,3,5,1,4] => [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,3,5,4,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,4,1,3,5] => [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[2,4,1,5,3] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[2,4,3,1,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,4,3,5,1] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,4,5,3,1] => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,5,3,4,1] => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,5,4,3,1] => [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,5,4,2] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,2,5,4,1] => [3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,4,5,2,1] => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,5,1,4,2] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,5,2,4,1] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,5,4,1,2] => [5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,5,4,2,1] => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,1,5,3,2] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,2,5,3,1] => [2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,3,5,2,1] => [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,5,1,3,2] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,5,2,3,1] => [4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,5,3,1,2] => [4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,5,3,2,1] => [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,1,2,4,3] => [5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,1,3,4,2] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,1,4,2,3] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,1,4,3,2] => [5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,2,1,4,3] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,2,3,4,1] => [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,2,4,1,3] => [5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,2,4,3,1] => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,3,1,4,2] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,3,2,4,1] => [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,3,4,1,2] => [3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,3,4,2,1] => [3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,1,2,3] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,1,3,2] => [5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,2,1,3] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,2,3,1] => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,3,1,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,3,4,6,5,1] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,3,5,6,4,1] => [5,6,2,3,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,3,6,4,5,1] => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,3,6,5,4,1] => [6,5,2,3,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,4,3,6,5,1] => [4,6,2,3,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,4,5,6,3,1] => [4,5,6,2,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,4,6,3,5,1] => [6,4,2,3,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 0 + 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St001399
(load all 3 compositions to match this statistic)
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001399: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
[2,3,1] => 00 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,1,2] => 00 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,2,1] => 00 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,3,4,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,4,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,4,3,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,1,4,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 1 + 1
[2,3,1,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,3,4,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,4,1,3] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,4,3,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,2,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,1,4,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,2,1,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,2,4,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,4,1,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,4,2,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,1,2,3] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,1,3,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,2,1,3] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,2,3,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,3,1,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,3,2,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,4,5,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[1,2,5,3,4] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[1,2,5,4,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[1,3,2,5,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 1 + 1
[1,3,4,2,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 + 1
[1,3,4,5,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,3,5,2,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,3,5,4,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,4,2,3,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 + 1
[1,4,2,5,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,4,3,2,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 + 1
[1,4,3,5,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,4,5,2,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,4,5,3,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,5,2,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,5,2,4,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,5,3,2,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,5,3,4,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,5,4,2,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,5,4,3,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[2,1,3,5,4] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 1 + 1
[2,1,4,3,5] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 1 + 1
[2,1,4,5,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1 + 1
[2,1,5,3,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1 + 1
[2,1,5,4,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1 + 1
[2,3,1,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[2,3,1,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1 + 1
[2,3,4,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[2,3,4,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,3,5,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,3,5,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,4,1,3,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[2,4,1,5,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,4,3,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[2,4,3,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,4,5,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,4,5,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,5,1,3,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,5,1,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,5,3,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,5,3,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,5,4,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,5,4,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,2,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[3,1,2,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1 + 1
[3,1,4,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[3,1,4,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,5,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,5,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,2,1,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[3,2,1,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1 + 1
[3,2,4,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[3,2,4,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,2,5,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,2,5,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,1,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[3,4,1,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,2,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[3,4,2,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,5,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,5,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,5,1,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,5,1,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,5,2,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,5,2,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,1,2,3,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[4,1,3,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[4,2,1,3,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[4,2,3,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[4,3,1,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[4,3,2,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,2,3,5,6,4] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 0 + 1
[1,2,3,6,4,5] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 0 + 1
[1,2,3,6,5,4] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 0 + 1
[1,2,4,3,6,5] => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 1 + 1
[1,2,4,5,3,6] => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 0 + 1
[1,2,4,5,6,3] => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 0 + 1
Description
The distinguishing number of a poset. This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring. See also [[St000469]], which is the same concept for graphs.
Matching statistic: St001964
(load all 22 compositions to match this statistic)
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001964: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 12%
Values
[2,3,1] => 00 => ([(0,2),(2,1)],3)
 => 0
[3,1,2] => 00 => ([(0,2),(2,1)],3)
 => 0
[3,2,1] => 00 => ([(0,2),(2,1)],3)
 => 0
[1,3,4,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[1,4,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[1,4,3,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[2,1,4,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1
[2,3,1,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[2,3,4,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[2,4,1,3] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[2,4,3,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,1,2,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[3,1,4,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,2,1,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[3,2,4,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,4,1,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,4,2,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,1,2,3] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,1,3,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,2,1,3] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,2,3,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,3,1,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,3,2,1] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,2,4,5,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[1,2,5,3,4] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[1,2,5,4,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[1,3,2,5,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 1
[1,3,4,2,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[1,3,4,5,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,3,5,2,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,3,5,4,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,4,2,3,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[1,4,2,5,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,4,3,2,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[1,4,3,5,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,4,5,2,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,4,5,3,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,5,2,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,5,2,4,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,5,3,2,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,5,3,4,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,5,4,2,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,5,4,3,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,1,3,5,4] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 1
[2,1,4,3,5] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 1
[2,1,4,5,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1
[2,1,5,3,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1
[2,1,5,4,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1
[2,3,1,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[2,3,1,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1
[2,3,4,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,3,4,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,3,5,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,3,5,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,4,1,3,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,4,1,5,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,4,3,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,4,3,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,4,5,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,4,5,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,5,1,3,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,5,1,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,5,3,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,5,3,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,5,4,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,5,4,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,1,2,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[3,1,2,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1
[3,1,4,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[3,1,4,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,1,5,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,1,5,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,2,1,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[3,2,1,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 1
[3,2,4,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[3,2,4,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,2,5,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,2,5,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,4,1,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[3,4,1,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,4,2,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[3,4,2,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,4,5,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,4,5,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,5,1,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,5,1,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,5,2,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,5,2,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,5,4,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,5,4,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[4,1,2,3,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[4,1,2,5,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[4,1,3,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[4,1,3,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[4,1,5,2,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[4,1,5,3,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[4,2,1,3,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[4,2,1,5,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[4,2,3,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[4,3,1,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
Description
The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000068
(load all 31 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 12%
Values
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 0 + 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,5,2,3,4] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[1,5,2,4,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 1 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 0 + 1
[2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[2,3,5,1,4] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ? = 0 + 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 0 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 0 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ? = 0 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[2,4,3,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 0 + 1
[2,4,5,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 0 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ? = 0 + 1
[2,5,1,4,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ? = 0 + 1
[2,5,3,1,4] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 0 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,5,4,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 0 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 1 + 1
[3,1,4,2,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 0 + 1
[3,1,4,5,2] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[3,1,5,2,4] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ? = 0 + 1
[3,1,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 0 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 1 + 1
[3,2,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 0 + 1
[3,2,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[3,2,5,1,4] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ? = 0 + 1
[3,2,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 0 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[3,4,1,5,2] => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 0 + 1
[3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[3,4,2,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 0 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,5,1,2,4] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 0 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,5,2,1,4] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 0 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,1,2,3,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 0 + 1
[4,1,2,5,3] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[4,1,3,2,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 0 + 1
[4,1,3,5,2] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[4,1,5,2,3] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 0 + 1
[4,1,5,3,2] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 0 + 1
[4,2,1,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
Description
The number of minimal elements in a poset.
The following 109 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001490The number of connected components of a skew partition. 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. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001845The number of join irreducibles minus the rank of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St001851The number of Hecke atoms of a signed permutation. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001779The order of promotion on the set of linear extensions of a poset. St000327The number of cover relations in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000634The number of endomorphisms of a poset. St001884The number of borders of a binary word. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000407The number of occurrences of the pattern 2143 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001768The number of reduced words of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000717The number of ordinal summands of a poset. 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 $eA$. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001970The signature of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001654The monophonic hull number of a graph. St001716The 1-improper chromatic number of a graph.