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Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000366
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 0
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [1,2] => 0
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 0
[+,-,+] => [1,3,2] => 0
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 0
[-,+,-] => [2,1,3] => 0
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,2,3] => 0
[-,3,2] => [2,1,3] => 0
[2,1,+] => [1,3,2] => 0
[2,1,-] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,+,1] => [2,1,3] => 0
[3,-,1] => [1,3,2] => 0
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 0
[+,-,+,+] => [1,3,4,2] => 0
[+,+,-,+] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 0
[-,+,-,+] => [2,4,1,3] => 0
[-,+,+,-] => [2,3,1,4] => 0
[+,-,-,+] => [1,4,2,3] => 0
[+,-,+,-] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 0
[-,-,+,-] => [3,1,2,4] => 0
[-,+,-,-] => [2,1,3,4] => 0
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => 0
[+,-,4,3] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => 0
[+,3,2,+] => [1,2,4,3] => 0
[-,3,2,+] => [2,4,1,3] => 0
[+,3,2,-] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => 0
[+,3,4,2] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => 0
[+,4,2,3] => [1,2,3,4] => 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000365
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [2,1] => 0
[-,+] => [2,1] => [1,2] => 0
[+,-] => [1,2] => [2,1] => 0
[-,-] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[+,+,+] => [1,2,3] => [3,2,1] => 0
[-,+,+] => [2,3,1] => [2,1,3] => 0
[+,-,+] => [1,3,2] => [3,1,2] => 0
[+,+,-] => [1,2,3] => [3,2,1] => 0
[-,-,+] => [3,1,2] => [1,3,2] => 0
[-,+,-] => [2,1,3] => [2,3,1] => 0
[+,-,-] => [1,2,3] => [3,2,1] => 0
[-,-,-] => [1,2,3] => [3,2,1] => 0
[+,3,2] => [1,2,3] => [3,2,1] => 0
[-,3,2] => [2,1,3] => [2,3,1] => 0
[2,1,+] => [1,3,2] => [3,1,2] => 0
[2,1,-] => [1,2,3] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => 0
[3,+,1] => [2,1,3] => [2,3,1] => 0
[3,-,1] => [1,3,2] => [3,1,2] => 0
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => 0
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => 0
[+,-,+,+] => [1,3,4,2] => [4,2,1,3] => 0
[+,+,-,+] => [1,2,4,3] => [4,3,1,2] => 0
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => 0
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 0
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => 0
[+,-,-,+] => [1,4,2,3] => [4,1,3,2] => 0
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => 0
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => 0
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => 0
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => 0
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[+,+,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => 0
[+,-,4,3] => [1,3,2,4] => [4,2,3,1] => 0
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => 0
[+,3,2,+] => [1,2,4,3] => [4,3,1,2] => 0
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => 0
[+,3,2,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => 0
[+,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => 0
[+,4,2,3] => [1,2,3,4] => [4,3,2,1] => 0
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St000731
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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] => [3,2,1] => 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] => [4,2,3,1] => 0
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => 0
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [4,1,3,2] => 0
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => 0
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => 0
[+,-,-,+] => [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] => [3,2,1,4] => 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] => [4,2,1,3] => 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
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000732
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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] => [3,2,1] => 0
[+,-,+] => [1,3,2] => [1,3,2] => 0
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [2,3,1] => 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] => [4,2,3,1] => 0
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => 0
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [2,4,3,1] => 0
[-,+,-,+] => [2,4,1,3] => [3,2,4,1] => 0
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => 0
[+,-,-,+] => [1,4,2,3] => [1,3,4,2] => 0
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [2,3,4,1] => 0
[-,-,+,-] => [3,1,2,4] => [2,3,1,4] => 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] => [3,2,1,4] => 0
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => [2,3,1,4] => 0
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => 0
[-,3,2,+] => [2,4,1,3] => [3,2,4,1] => 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] => [3,2,1,4] => 0
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => 0
Description
The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St001964
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => ([],1)
=> 0
[-] => [1] => [1] => ([],1)
=> 0
[+,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[-,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[+,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[-,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 0
[+,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[-,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,+] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,-] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,1] => [2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [3,1,2] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,+,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,-,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[+,3,2,+] => [1,3,2,4] => [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
[-,3,2,+] => [1,3,2,4] => [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
[+,3,2,-] => [1,3,2,4] => [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
[-,3,2,-] => [1,3,2,4] => [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
[+,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[-,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[+,4,2,3] => [1,4,2,3] => [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
[-,4,2,3] => [1,4,2,3] => [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
[+,4,+,2] => [1,4,3,2] => [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
[-,4,+,2] => [1,4,3,2] => [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)
=> ? = 1
[+,4,-,2] => [1,4,3,2] => [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
[-,4,-,2] => [1,4,3,2] => [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
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 0
[2,3,1,+] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,3,1,-] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,3,4,1] => [2,3,4,1] => [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
[2,4,1,3] => [2,4,1,3] => [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
[2,4,+,1] => [2,4,3,1] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[2,4,-,1] => [2,4,3,1] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,1,2,+] => [3,1,2,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,1,2,-] => [3,1,2,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 0
[3,+,1,+] => [3,2,1,4] => [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
[3,-,1,+] => [3,2,1,4] => [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)
=> ? = 1
[3,+,1,-] => [3,2,1,4] => [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
[3,-,1,-] => [3,2,1,4] => [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
[3,+,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,-,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,1,2] => [3,4,1,2] => [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
[3,4,2,1] => [3,4,2,1] => [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
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,1,+,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,1,-,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,+,1,3] => [4,2,1,3] => [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
[4,-,1,3] => [4,2,1,3] => [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
[4,+,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,-,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,+,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,-,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,3,1,2] => [4,3,1,2] => [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)
=> ? = 0
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[+,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(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: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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: 33%
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: 33%
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)
=> ? = 0 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[-] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[+,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[-,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[+,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[-,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[+,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,+] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [3,1,2] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,+,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,3,2,+] => [1,3,2,4] => [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
[-,3,2,+] => [1,3,2,4] => [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
[+,3,2,-] => [1,3,2,4] => [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
[-,3,2,-] => [1,3,2,4] => [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
[+,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,4,2,3] => [1,4,2,3] => [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
[-,4,2,3] => [1,4,2,3] => [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
[+,4,+,2] => [1,4,3,2] => [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
[-,4,+,2] => [1,4,3,2] => [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)
=> ? = 1 + 1
[+,4,-,2] => [1,4,3,2] => [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
[-,4,-,2] => [1,4,3,2] => [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
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 0 + 1
[2,3,1,+] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,3,1,-] => [2,3,1,4] => [3,2,1,4] => ([(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] => [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
[2,4,1,3] => [2,4,1,3] => [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
[2,4,+,1] => [2,4,3,1] => [4,2,1,3] => ([(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] => [4,2,1,3] => ([(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] => [2,3,1,4] => ([(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] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => ([(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] => [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
[3,-,1,+] => [3,2,1,4] => [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)
=> ? = 1 + 1
[3,+,1,-] => [3,2,1,4] => [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
[3,-,1,-] => [3,2,1,4] => [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
[3,+,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,-,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [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
[3,4,2,1] => [3,4,2,1] => [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
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,1,+,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,1,-,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,+,1,3] => [4,2,1,3] => [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
[4,-,1,3] => [4,2,1,3] => [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
[4,+,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,-,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,+,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,-,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,3,1,2] => [4,3,1,2] => [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)
=> ? = 0 + 1
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(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] => [1,2,3,5,4] => [1,2,3,5,4] => ([(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] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001890
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => ([],1)
=> ? = 0 + 1
[-] => [1] => [1] => ([],1)
=> ? = 0 + 1
[+,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[-,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[+,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[-,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[+,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,+] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [3,1,2] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,+,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,3,2,+] => [1,3,2,4] => [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
[-,3,2,+] => [1,3,2,4] => [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
[+,3,2,-] => [1,3,2,4] => [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
[-,3,2,-] => [1,3,2,4] => [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
[+,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,4,2,3] => [1,4,2,3] => [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
[-,4,2,3] => [1,4,2,3] => [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
[+,4,+,2] => [1,4,3,2] => [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
[-,4,+,2] => [1,4,3,2] => [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)
=> ? = 1 + 1
[+,4,-,2] => [1,4,3,2] => [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
[-,4,-,2] => [1,4,3,2] => [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
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 0 + 1
[2,3,1,+] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,3,1,-] => [2,3,1,4] => [3,2,1,4] => ([(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] => [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
[2,4,1,3] => [2,4,1,3] => [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
[2,4,+,1] => [2,4,3,1] => [4,2,1,3] => ([(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] => [4,2,1,3] => ([(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] => [2,3,1,4] => ([(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] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => ([(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] => [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
[3,-,1,+] => [3,2,1,4] => [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)
=> ? = 1 + 1
[3,+,1,-] => [3,2,1,4] => [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
[3,-,1,-] => [3,2,1,4] => [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
[3,+,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,-,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [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
[3,4,2,1] => [3,4,2,1] => [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
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,1,+,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,1,-,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,+,1,3] => [4,2,1,3] => [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
[4,-,1,3] => [4,2,1,3] => [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
[4,+,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,-,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,+,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,-,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,3,1,2] => [4,3,1,2] => [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)
=> ? = 0 + 1
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 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: St001630
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => ([(0,1)],2)
=> ? = 0 + 2
[-] => [1] => [1] => ([(0,1)],2)
=> ? = 0 + 2
[+,+] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[-,+] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[+,-] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[-,-] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1] => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[+,+,+] => [1,2,3] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[-,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[2,1,+] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[2,1,-] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[2,3,1] => [2,3,1] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[3,1,2] => [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] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[3,-,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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
[-,+,4,3] => [1,2,4,3] => [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
[+,-,4,3] => [1,2,4,3] => [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
[-,-,4,3] => [1,2,4,3] => [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,+] => [1,3,2,4] => [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
[-,3,2,+] => [1,3,2,4] => [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
[+,3,2,-] => [1,3,2,4] => [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
[-,3,2,-] => [1,3,2,4] => [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] => [1,3,4,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[-,3,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[+,4,2,3] => [1,4,2,3] => [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
[-,4,2,3] => [1,4,2,3] => [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] => [1,4,3,2] => [4,3,1,2] => ([(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,+,2] => [1,4,3,2] => [4,3,1,2] => ([(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)
=> ? = 1 + 2
[+,4,-,2] => [1,4,3,2] => [4,3,1,2] => ([(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,-,2] => [1,4,3,2] => [4,3,1,2] => ([(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,+,+] => [2,1,3,4] => [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,-,+] => [2,1,3,4] => [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,+,-] => [2,1,3,4] => [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,-,-] => [2,1,3,4] => [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] => [2,1,4,3] => [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
[2,3,1,+] => [2,3,1,4] => [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
[2,3,1,-] => [2,3,1,4] => [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
[2,3,4,1] => [2,3,4,1] => [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
[2,4,1,3] => [2,4,1,3] => [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
[2,4,+,1] => [2,4,3,1] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 2
[2,4,-,1] => [2,4,3,1] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 2
[3,1,2,+] => [3,1,2,4] => [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)
=> ? = 0 + 2
[3,1,2,-] => [3,1,2,4] => [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)
=> ? = 0 + 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 2
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => ([(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,+] => [3,2,1,4] => [3,2,1,4] => ([(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)
=> ? = 1 + 2
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => ([(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,-] => [3,2,1,4] => [3,2,1,4] => ([(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] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[+,4,5,2,3] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[-,4,5,2,3] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,+,1,4] => [2,5,3,1,4] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,-,1,4] => [2,5,3,1,4] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,+,+,1] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,-,+,1] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,+,-,1] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,-,-,1] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[3,1,4,5,2] => [3,1,4,5,2] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[4,+,1,5,3] => [4,2,1,5,3] => [2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[4,-,1,5,3] => [4,2,1,5,3] => [2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[4,+,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[4,-,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => ([(0,1)],2)
=> ? = 0 + 2
[-] => [1] => [1] => ([(0,1)],2)
=> ? = 0 + 2
[+,+] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[-,+] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[+,-] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[-,-] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1] => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[+,+,+] => [1,2,3] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[-,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[2,1,+] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[2,1,-] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[2,3,1] => [2,3,1] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[3,1,2] => [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] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[3,-,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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
[-,+,4,3] => [1,2,4,3] => [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
[+,-,4,3] => [1,2,4,3] => [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
[-,-,4,3] => [1,2,4,3] => [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,+] => [1,3,2,4] => [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
[-,3,2,+] => [1,3,2,4] => [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
[+,3,2,-] => [1,3,2,4] => [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
[-,3,2,-] => [1,3,2,4] => [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] => [1,3,4,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[-,3,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[+,4,2,3] => [1,4,2,3] => [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
[-,4,2,3] => [1,4,2,3] => [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] => [1,4,3,2] => [4,3,1,2] => ([(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,+,2] => [1,4,3,2] => [4,3,1,2] => ([(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)
=> ? = 1 + 2
[+,4,-,2] => [1,4,3,2] => [4,3,1,2] => ([(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,-,2] => [1,4,3,2] => [4,3,1,2] => ([(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,+,+] => [2,1,3,4] => [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,-,+] => [2,1,3,4] => [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,+,-] => [2,1,3,4] => [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,-,-] => [2,1,3,4] => [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] => [2,1,4,3] => [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
[2,3,1,+] => [2,3,1,4] => [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
[2,3,1,-] => [2,3,1,4] => [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
[2,3,4,1] => [2,3,4,1] => [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
[2,4,1,3] => [2,4,1,3] => [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
[2,4,+,1] => [2,4,3,1] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 2
[2,4,-,1] => [2,4,3,1] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 2
[3,1,2,+] => [3,1,2,4] => [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)
=> ? = 0 + 2
[3,1,2,-] => [3,1,2,4] => [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)
=> ? = 0 + 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 2
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => ([(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,+] => [3,2,1,4] => [3,2,1,4] => ([(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)
=> ? = 1 + 2
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => ([(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,-] => [3,2,1,4] => [3,2,1,4] => ([(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] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[+,4,5,2,3] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[-,4,5,2,3] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,+,1,4] => [2,5,3,1,4] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,-,1,4] => [2,5,3,1,4] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,+,+,1] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,-,+,1] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,+,-,1] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[2,5,-,-,1] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[3,1,4,5,2] => [3,1,4,5,2] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[4,+,1,5,3] => [4,2,1,5,3] => [2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[4,-,1,5,3] => [4,2,1,5,3] => [2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[4,+,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[4,-,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 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.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000907The number of maximal antichains of minimal length in a poset. St001857The number of edges in the reduced word graph of a signed permutation. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
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