Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000405
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[+,-,+] => [1,3,2] => [3,1,2] => [3,2,1] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [3,1,2] => [3,2,1] => 0
[-,3,2] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[2,1,+] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[2,1,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,+,1] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,-,1] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[+,-,+,+] => [1,3,4,2] => [2,4,1,3] => [4,3,1,2] => 0
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,3,2,1] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [4,2,1,3] => 0
[-,+,-,+] => [2,4,1,3] => [1,3,4,2] => [1,4,2,3] => 0
[-,+,+,-] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[+,-,-,+] => [1,4,2,3] => [3,4,1,2] => [3,1,4,2] => 0
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[-,-,+,-] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [4,1,2,3] => [4,3,2,1] => 0
[-,+,4,3] => [2,4,1,3] => [1,3,4,2] => [1,4,2,3] => 0
[+,-,4,3] => [1,4,2,3] => [3,4,1,2] => [3,1,4,2] => 0
[-,-,4,3] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[+,3,2,+] => [1,3,4,2] => [2,4,1,3] => [4,3,1,2] => 0
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => [4,2,1,3] => 0
[+,3,2,-] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[-,3,2,-] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[+,3,4,2] => [1,4,2,3] => [3,4,1,2] => [3,1,4,2] => 0
[-,3,4,2] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[+,4,2,3] => [1,3,4,2] => [2,4,1,3] => [4,3,1,2] => 0
Description
The number of occurrences of the pattern 1324 in a permutation. There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that $$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St000456
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => [1] => ([],1)
 => ? = 0 + 1
[-] => [1] => [1] => ([],1)
 => ? = 0 + 1
[+,+] => [1,2] => [2] => ([],2)
 => ? = 0 + 1
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[+,-] => [1,2] => [2] => ([],2)
 => ? = 0 + 1
[-,-] => [1,2] => [2] => ([],2)
 => ? = 0 + 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[+,+,+] => [1,2,3] => [3] => ([],3)
 => ? = 0 + 1
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[+,+,-] => [1,2,3] => [3] => ([],3)
 => ? = 0 + 1
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 0 + 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 0 + 1
[+,-,-] => [1,2,3] => [3] => ([],3)
 => ? = 0 + 1
[-,-,-] => [1,2,3] => [3] => ([],3)
 => ? = 0 + 1
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 0 + 1
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 0 + 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 0 + 1
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,-,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 0 + 1
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
 => ? = 0 + 1
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [4] => ([],4)
 => ? = 0 + 1
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? = 0 + 1
[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[+,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? = 0 + 1
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? = 0 + 1
[+,+,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,+,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[-,-,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[+,3,2,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,3,2,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[+,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[-,3,4,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[+,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,4,2,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[+,4,+,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,4,+,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[-,4,-,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[2,1,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[2,4,1,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,4,+,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,4,-,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0 + 1
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,+,1,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,1,2,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,1,+,2] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,+,1,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,+,+,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,-,+,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,+,-,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,+,+,-,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,+,+,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,+,4,3,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,+,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,+,5,+,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,3,2,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,4,2,3,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,4,+,2,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,5,2,+,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,5,+,2,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[+,5,+,+,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,1,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1,2,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,+,1,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,2,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,+,2,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,+,1,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,+,+,1,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[5,1,2,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[5,1,+,2,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[5,1,+,+,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[5,+,1,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[5,+,1,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 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: St000181
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => [1] => ([],1)
 => 1 = 0 + 1
[-] => [1] => [1] => ([],1)
 => 1 = 0 + 1
[+,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[-,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[+,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[-,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[+,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[-,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1,+] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,-] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,+,1] => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,-,1] => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[+,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,+,4,3] => [1,2,4,3] => [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)
 => ? = 0 + 1
[-,+,4,3] => [1,2,4,3] => [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)
 => ? = 0 + 1
[+,-,4,3] => [1,2,4,3] => [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)
 => ? = 0 + 1
[-,-,4,3] => [1,2,4,3] => [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)
 => ? = 0 + 1
[+,3,2,+] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[-,3,2,+] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[+,3,2,-] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[-,3,2,-] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[+,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[-,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[+,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,4,+,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,+,+] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,1,-,+] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,1,+,-] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1 + 1
[2,1,-,-] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[2,3,1,+] => [2,3,1,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[2,3,1,-] => [2,3,1,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,4,+,1] => [2,4,3,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,4,-,1] => [2,4,3,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[3,1,2,+] => [3,1,2,4] => [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
[3,1,2,-] => [3,1,2,4] => [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)
 => ? = 1 + 1
[3,1,4,2] => [3,1,4,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[3,+,1,+] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,-,1,+] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,+,1,-] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1 + 1
[3,-,1,-] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,+,4,1] => [3,2,4,1] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[3,-,4,1] => [3,2,4,1] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[+,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001630
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => ([(0,1)],2)
 => ? = 0 + 2
[-] => [1] => ([(0,1)],2)
 => ? = 0 + 2
[+,+] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[-,+] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[+,-] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[-,-] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[+,+,+] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[-,+,+] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[+,-,+] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[+,+,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[-,-,+] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[-,+,-] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[+,-,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[-,-,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[+,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[-,3,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,+] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,-] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,3,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,2] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,+,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,-,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[-,+,+,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[+,-,+,+] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[+,+,-,+] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[+,+,+,-] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[-,-,+,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[-,+,-,+] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[-,+,+,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[+,-,-,+] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[+,-,+,-] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[+,+,-,-] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[-,-,-,+] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[-,-,+,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,+,-,-] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[+,-,-,-] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[-,-,-,-] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[+,+,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[-,+,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[+,-,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,-,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[+,3,2,+] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,3,2,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[+,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[-,3,2,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[+,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,3,4,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[+,4,2,3] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,4,2,3] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[+,4,+,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,4,+,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[+,4,-,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,4,-,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[2,1,+,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[2,1,-,+] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,1,+,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[2,1,-,-] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,3,1,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,3,1,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[2,3,4,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[2,4,1,3] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,4,+,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,4,-,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[3,1,2,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[3,1,2,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[3,1,4,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[3,+,1,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[3,-,1,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[3,+,1,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[3,-,1,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[3,+,4,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[3,-,4,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[3,4,1,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[3,4,2,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[4,1,-,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[4,+,-,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => ([(0,1)],2)
 => ? = 0 + 2
[-] => [1] => ([(0,1)],2)
 => ? = 0 + 2
[+,+] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[-,+] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[+,-] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[-,-] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[+,+,+] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[-,+,+] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[+,-,+] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[+,+,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[-,-,+] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[-,+,-] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[+,-,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[-,-,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[+,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[-,3,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,+] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,-] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,3,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,2] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,+,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,-,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[-,+,+,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[+,-,+,+] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[+,+,-,+] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[+,+,+,-] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[-,-,+,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[-,+,-,+] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[-,+,+,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[+,-,-,+] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[+,-,+,-] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[+,+,-,-] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[-,-,-,+] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[-,-,+,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,+,-,-] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[+,-,-,-] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[-,-,-,-] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
[+,+,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[-,+,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[+,-,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,-,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[+,3,2,+] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,3,2,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[+,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[-,3,2,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[+,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,3,4,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[+,4,2,3] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,4,2,3] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[+,4,+,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,4,+,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[+,4,-,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[-,4,-,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[2,1,+,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[2,1,-,+] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,1,+,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[2,1,-,-] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 0 + 2
[2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,3,1,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,3,1,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[2,3,4,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[2,4,1,3] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,4,+,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,4,-,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[3,1,2,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[3,1,2,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[3,1,4,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[3,+,1,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[3,-,1,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[3,+,1,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
[3,-,1,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 0 + 2
[3,+,4,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[3,-,4,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 0 + 2
[3,4,1,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[3,4,2,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 2
[4,1,-,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[4,+,-,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001890
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => [1] => ([],1)
 => ? = 0 + 1
[-] => [1] => [1] => ([],1)
 => ? = 0 + 1
[+,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[-,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[+,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[-,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[+,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[-,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1,+] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,-] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,+,1] => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,-,1] => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[+,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,+,4,3] => [1,2,4,3] => [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)
 => ? = 0 + 1
[-,+,4,3] => [1,2,4,3] => [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)
 => ? = 0 + 1
[+,-,4,3] => [1,2,4,3] => [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)
 => ? = 0 + 1
[-,-,4,3] => [1,2,4,3] => [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)
 => ? = 0 + 1
[+,3,2,+] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[-,3,2,+] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[+,3,2,-] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[-,3,2,-] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[+,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[-,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[+,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,4,+,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,+,+] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,1,-,+] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,1,+,-] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1 + 1
[2,1,-,-] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[2,3,1,+] => [2,3,1,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[2,3,1,-] => [2,3,1,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,4,+,1] => [2,4,3,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[2,4,-,1] => [2,4,3,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[3,1,2,+] => [3,1,2,4] => [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
[3,1,2,-] => [3,1,2,4] => [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)
 => ? = 1 + 1
[3,1,4,2] => [3,1,4,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0 + 1
[3,+,1,+] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,-,1,+] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,+,1,-] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1 + 1
[3,-,1,-] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,+,4,1] => [3,2,4,1] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[3,-,4,1] => [3,2,4,1] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[+,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
Description
The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Matching statistic: St001857
(load all 20 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => 0
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [2,3,1] => 0
[+,-,+] => [1,3,2] => [1,3,2] => 0
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,1,2] => 0
[-,+,-] => [2,1,3] => [2,1,3] => 0
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,+] => [1,3,2] => [1,3,2] => 0
[2,1,-] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,1,3] => 0
[3,-,1] => [1,3,2] => [1,3,2] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ? = 0
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => ? = 0
[+,-,+,+] => [1,3,4,2] => [1,3,4,2] => ? = 0
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => ? = 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ? = 0
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => ? = 0
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => ? = 0
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => ? = 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => ? = 0
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => ? = 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ? = 0
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => ? = 0
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => ? = 0
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ? = 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ? = 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ? = 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => ? = 0
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => ? = 0
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => ? = 0
[-,-,4,3] => [3,1,2,4] => [3,1,2,4] => ? = 0
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => ? = 0
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => ? = 0
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => ? = 0
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => ? = 0
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => ? = 0
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => ? = 0
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => ? = 0
[-,4,2,3] => [2,3,1,4] => [2,3,1,4] => ? = 0
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => ? = 0
[-,4,+,2] => [3,2,1,4] => [3,2,1,4] => ? = 0
[+,4,-,2] => [1,2,4,3] => [1,2,4,3] => ? = 0
[-,4,-,2] => [2,1,4,3] => [2,1,4,3] => ? = 0
[2,1,+,+] => [1,3,4,2] => [1,3,4,2] => ? = 0
[2,1,-,+] => [1,4,2,3] => [1,4,2,3] => ? = 0
[2,1,+,-] => [1,3,2,4] => [1,3,2,4] => ? = 1
[2,1,-,-] => [1,2,3,4] => [1,2,3,4] => ? = 0
[2,1,4,3] => [1,3,2,4] => [1,3,2,4] => ? = 0
[2,3,1,+] => [1,4,2,3] => [1,4,2,3] => ? = 0
[2,3,1,-] => [1,2,3,4] => [1,2,3,4] => ? = 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ? = 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => ? = 0
[2,4,+,1] => [3,1,2,4] => [3,1,2,4] => ? = 0
[2,4,-,1] => [1,2,4,3] => [1,2,4,3] => ? = 0
[3,1,2,+] => [1,2,4,3] => [1,2,4,3] => ? = 0
[3,1,2,-] => [1,2,3,4] => [1,2,3,4] => ? = 1
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => ? = 0
[3,+,1,+] => [2,1,4,3] => [2,1,4,3] => ? = 0
[3,-,1,+] => [1,4,3,2] => [1,4,3,2] => ? = 0
[3,+,1,-] => [2,1,3,4] => [2,1,3,4] => ? = 1
[3,-,1,-] => [1,3,2,4] => [1,3,2,4] => ? = 0
Description
The number of edges in the reduced word graph of a signed permutation. The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St000084
(load all 2 compositions to match this statistic)
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => [.,.]
 => [[],[]]
 => 2 = 0 + 2
[-] => [1] => [.,.]
 => [[],[]]
 => 2 = 0 + 2
[+,+] => [1,2] => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[-,+] => [1,2] => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[+,-] => [1,2] => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[-,-] => [1,2] => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[2,1] => [2,1] => [[.,.],.]
 => [[[],[]],[]]
 => 2 = 0 + 2
[+,+,+] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[-,+,+] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[+,-,+] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[+,+,-] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[-,-,+] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[-,+,-] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[+,-,-] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[-,-,-] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[+,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2 = 0 + 2
[-,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2 = 0 + 2
[2,1,+] => [2,1,3] => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2 = 0 + 2
[2,1,-] => [2,1,3] => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2 = 0 + 2
[2,3,1] => [2,3,1] => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2 = 0 + 2
[3,1,2] => [3,1,2] => [[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 2 = 0 + 2
[3,+,1] => [3,2,1] => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2 = 0 + 2
[3,-,1] => [3,2,1] => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,-,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,+,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,-,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,+,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 1 + 2
[+,-,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,-,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,-,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,-,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => ? = 0 + 2
[-,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => ? = 0 + 2
[+,-,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => ? = 0 + 2
[-,-,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => ? = 0 + 2
[+,3,2,+] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[-,3,2,+] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[+,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[-,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[+,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[-,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[+,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => ? = 0 + 2
[-,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => ? = 0 + 2
[+,4,+,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => ? = 0 + 2
[-,4,+,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => ? = 0 + 2
[+,4,-,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => ? = 0 + 2
[-,4,-,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => ? = 0 + 2
[2,1,+,+] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,1,-,+] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,1,+,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 1 + 2
[2,1,-,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => ? = 0 + 2
[2,3,1,+] => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,3,1,-] => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,4,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => ? = 0 + 2
[2,4,+,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => ? = 0 + 2
[2,4,-,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => ? = 0 + 2
[3,1,2,+] => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => ? = 0 + 2
[3,1,2,-] => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => ? = 1 + 2
[3,1,4,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => ? = 0 + 2
[3,+,1,+] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => ? = 0 + 2
[3,-,1,+] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => ? = 0 + 2
[3,+,1,-] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => ? = 1 + 2
[3,-,1,-] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => ? = 0 + 2
Description
The number of subtrees.
Matching statistic: St000328
(load all 2 compositions to match this statistic)
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000328: Ordered trees ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => [.,.]
 => [[],[]]
 => 2 = 0 + 2
[-] => [1] => [.,.]
 => [[],[]]
 => 2 = 0 + 2
[+,+] => [1,2] => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[-,+] => [1,2] => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[+,-] => [1,2] => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[-,-] => [1,2] => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[2,1] => [2,1] => [[.,.],.]
 => [[[],[]],[]]
 => 2 = 0 + 2
[+,+,+] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[-,+,+] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[+,-,+] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[+,+,-] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[-,-,+] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[-,+,-] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[+,-,-] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[-,-,-] => [1,2,3] => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[+,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2 = 0 + 2
[-,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2 = 0 + 2
[2,1,+] => [2,1,3] => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2 = 0 + 2
[2,1,-] => [2,1,3] => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2 = 0 + 2
[2,3,1] => [2,3,1] => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2 = 0 + 2
[3,1,2] => [3,1,2] => [[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 2 = 0 + 2
[3,+,1] => [3,2,1] => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2 = 0 + 2
[3,-,1] => [3,2,1] => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,-,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,+,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,-,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,+,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 1 + 2
[+,-,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,-,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,-,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,-,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[-,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => ? = 0 + 2
[+,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => ? = 0 + 2
[-,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => ? = 0 + 2
[+,-,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => ? = 0 + 2
[-,-,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => ? = 0 + 2
[+,3,2,+] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[-,3,2,+] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[+,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[-,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[+,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[-,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => ? = 0 + 2
[+,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => ? = 0 + 2
[-,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => ? = 0 + 2
[+,4,+,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => ? = 0 + 2
[-,4,+,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => ? = 0 + 2
[+,4,-,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => ? = 0 + 2
[-,4,-,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => ? = 0 + 2
[2,1,+,+] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,1,-,+] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,1,+,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 1 + 2
[2,1,-,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => ? = 0 + 2
[2,3,1,+] => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,3,1,-] => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => ? = 0 + 2
[2,4,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => ? = 0 + 2
[2,4,+,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => ? = 0 + 2
[2,4,-,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => ? = 0 + 2
[3,1,2,+] => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => ? = 0 + 2
[3,1,2,-] => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => ? = 1 + 2
[3,1,4,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => ? = 0 + 2
[3,+,1,+] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => ? = 0 + 2
[3,-,1,+] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => ? = 0 + 2
[3,+,1,-] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => ? = 1 + 2
[3,-,1,-] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => ? = 0 + 2
Description
The maximum number of child nodes in a tree.
Matching statistic: St001926
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001926: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => [1] => [-1] => ? = 0
[-] => [1] => [1] => [-1] => ? = 0
[+,+] => [1,2] => [1,2] => [-1,-2] => 0
[-,+] => [1,2] => [1,2] => [-1,-2] => 0
[+,-] => [1,2] => [1,2] => [-1,-2] => 0
[-,-] => [1,2] => [1,2] => [-1,-2] => 0
[2,1] => [2,1] => [2,1] => [-2,-1] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[-,+,+] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[+,-,+] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[-,-,+] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[-,+,-] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[+,3,2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0
[-,3,2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0
[2,1,+] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0
[2,1,-] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0
[2,3,1] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 0
[3,+,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => ? = 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => ? = 0
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => ? = 0
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => ? = 0
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => ? = 0
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => ? = 0
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => ? = 0
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => ? = 0
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => ? = 0
[+,3,4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => ? = 0
[-,3,4,2] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => ? = 0
[+,4,2,3] => [1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => ? = 0
[-,4,2,3] => [1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => ? = 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => ? = 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => ? = 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => ? = 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => ? = 0
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => ? = 0
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => ? = 0
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => ? = 1
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => ? = 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => ? = 0
[2,3,1,+] => [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => ? = 0
[2,3,1,-] => [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => ? = 0
[2,3,4,1] => [2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => ? = 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => ? = 0
[2,4,+,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => ? = 0
[2,4,-,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => ? = 0
[3,1,2,+] => [3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => ? = 0
[3,1,2,-] => [3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => ? = 1
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => ? = 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => ? = 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => ? = 0
Description
Sparre Andersen's position of the maximum of a signed permutation. For $\pi$ a signed permutation of length $n$, first create the tuple $x = (x_1, \dots, x_n)$, where $x_i = c_{|\pi_1|} \operatorname{sgn}(\pi_{|\pi_1|}) + \cdots + c_{|\pi_i|} \operatorname{sgn}(\pi_{|\pi_i|})$ and $(c_1, \dots ,c_n) = (1, 2, \dots, 2^{n-1})$. The actual value of the c-tuple for Andersen's statistic does not matter so long as no sums or differences of any subset of the $c_i$'s is zero. The choice of powers of $2$ is just a convenient choice. This returns the largest position of the maximum value in the $x$-tuple. This is related to the ''discrete arcsine distribution''. The number of signed permutations with value equal to $k$ is given by $\binom{2k}{k} \binom{2n-2k}{n-k} \frac{n!}{2^n}$. This statistic is equidistributed with Sparre Andersen's `Number of Positives' statistic.
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St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.