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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000162
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
St000162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000162: 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] => 1
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 1
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => 1
[+,-,4,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => 1
Description
The number of nontrivial cycles in the cycle decomposition of a permutation.
This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St000996
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000996: 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] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,1,2] => 1
[+,-,+] => [1,3,2] => [1,3,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,2,1] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => [3,2,1] => 1
[2,1,+] => [2,3,1] => [3,1,2] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,2,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => 1
[3,+,1] => [2,3,1] => [3,1,2] => 1
[3,-,1] => [3,1,2] => [3,2,1] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => 1
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => 2
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => 1
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => 1
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => [4,3,1,2] => 1
[+,-,4,3] => [1,4,2,3] => [1,4,3,2] => 1
[-,-,4,3] => [4,1,2,3] => [4,3,2,1] => 1
[+,3,2,+] => [1,3,4,2] => [1,4,2,3] => 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [3,2,1,4] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,3,2] => 1
[-,3,4,2] => [4,1,2,3] => [4,3,2,1] => 1
[+,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001280
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1]
=> 0
[-] => [1] => [1]
=> 0
[+,+] => [1,2] => [1,1]
=> 0
[-,+] => [2,1] => [2]
=> 1
[+,-] => [1,2] => [1,1]
=> 0
[-,-] => [1,2] => [1,1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[+,+,+] => [1,2,3] => [1,1,1]
=> 0
[-,+,+] => [2,3,1] => [3]
=> 1
[+,-,+] => [1,3,2] => [2,1]
=> 1
[+,+,-] => [1,2,3] => [1,1,1]
=> 0
[-,-,+] => [3,1,2] => [3]
=> 1
[-,+,-] => [2,1,3] => [2,1]
=> 1
[+,-,-] => [1,2,3] => [1,1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> 1
[-,3,2] => [3,1,2] => [3]
=> 1
[2,1,+] => [2,3,1] => [3]
=> 1
[2,1,-] => [2,1,3] => [2,1]
=> 1
[2,3,1] => [3,1,2] => [3]
=> 1
[3,1,2] => [2,3,1] => [3]
=> 1
[3,+,1] => [2,3,1] => [3]
=> 1
[3,-,1] => [3,1,2] => [3]
=> 1
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> 0
[-,+,+,+] => [2,3,4,1] => [4]
=> 1
[+,-,+,+] => [1,3,4,2] => [3,1]
=> 1
[+,+,-,+] => [1,2,4,3] => [2,1,1]
=> 1
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> 0
[-,-,+,+] => [3,4,1,2] => [2,2]
=> 2
[-,+,-,+] => [2,4,1,3] => [4]
=> 1
[-,+,+,-] => [2,3,1,4] => [3,1]
=> 1
[+,-,-,+] => [1,4,2,3] => [3,1]
=> 1
[+,-,+,-] => [1,3,2,4] => [2,1,1]
=> 1
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> 0
[-,-,-,+] => [4,1,2,3] => [4]
=> 1
[-,-,+,-] => [3,1,2,4] => [3,1]
=> 1
[-,+,-,-] => [2,1,3,4] => [2,1,1]
=> 1
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[-,+,4,3] => [2,4,1,3] => [4]
=> 1
[+,-,4,3] => [1,4,2,3] => [3,1]
=> 1
[-,-,4,3] => [4,1,2,3] => [4]
=> 1
[+,3,2,+] => [1,3,4,2] => [3,1]
=> 1
[-,3,2,+] => [3,4,1,2] => [2,2]
=> 2
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> 1
[-,3,2,-] => [3,1,2,4] => [3,1]
=> 1
[+,3,4,2] => [1,4,2,3] => [3,1]
=> 1
[-,3,4,2] => [4,1,2,3] => [4]
=> 1
[+,4,2,3] => [1,3,4,2] => [3,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001737
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001737: 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] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,1,2] => 1
[+,-,+] => [1,3,2] => [1,3,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,2,1] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => [3,2,1] => 1
[2,1,+] => [2,3,1] => [3,1,2] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,2,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => 1
[3,+,1] => [2,3,1] => [3,1,2] => 1
[3,-,1] => [3,1,2] => [3,2,1] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => 1
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => 2
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => 1
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => 1
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => [4,3,1,2] => 1
[+,-,4,3] => [1,4,2,3] => [1,4,3,2] => 1
[-,-,4,3] => [4,1,2,3] => [4,3,2,1] => 1
[+,3,2,+] => [1,3,4,2] => [1,4,2,3] => 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [3,2,1,4] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,3,2] => 1
[-,3,4,2] => [4,1,2,3] => [4,3,2,1] => 1
[+,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St000374
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000374: 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] => 1
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,1,2] => [3,2,1] => 1
[+,-,+] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,2,1] => [2,3,1] => 1
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [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] => 1
[-,3,2] => [3,1,2] => [3,2,1] => [2,3,1] => 1
[2,1,+] => [2,3,1] => [3,1,2] => [3,2,1] => 1
[2,1,-] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,2,1] => [2,3,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => [3,2,1] => 1
[3,+,1] => [2,3,1] => [3,1,2] => [3,2,1] => 1
[3,-,1] => [3,1,2] => [3,2,1] => [2,3,1] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 1
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[+,+,+,-] => [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] => 2
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => [4,2,3,1] => 1
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => [1,3,4,2] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => [3,2,4,1] => 1
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => [2,3,1,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 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
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => [4,3,1,2] => [4,2,3,1] => 1
[+,-,4,3] => [1,4,2,3] => [1,4,3,2] => [1,3,4,2] => 1
[-,-,4,3] => [4,1,2,3] => [4,3,2,1] => [3,2,4,1] => 1
[+,3,2,+] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => [4,2,1,3] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [3,2,1,4] => [2,3,1,4] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,3,2] => [1,3,4,2] => 1
[-,3,4,2] => [4,1,2,3] => [4,3,2,1] => [3,2,4,1] => 1
[+,4,2,3] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000251
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => {{1}}
=> ? ∊ {0,0}
[-] => [1] => {{1}}
=> ? ∊ {0,0}
[+,+] => [1,2] => {{1},{2}}
=> 0
[-,+] => [2,1] => {{1,2}}
=> 1
[+,-] => [1,2] => {{1},{2}}
=> 0
[-,-] => [1,2] => {{1},{2}}
=> 0
[2,1] => [2,1] => {{1,2}}
=> 1
[+,+,+] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,+,+] => [2,3,1] => {{1,2,3}}
=> 1
[+,-,+] => [1,3,2] => {{1},{2,3}}
=> 1
[+,+,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,-,+] => [3,1,2] => {{1,2,3}}
=> 1
[-,+,-] => [2,1,3] => {{1,2},{3}}
=> 1
[+,-,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,-,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[+,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[-,3,2] => [3,1,2] => {{1,2,3}}
=> 1
[2,1,+] => [2,3,1] => {{1,2,3}}
=> 1
[2,1,-] => [2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 1
[3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[3,+,1] => [2,3,1] => {{1,2,3}}
=> 1
[3,-,1] => [3,1,2] => {{1,2,3}}
=> 1
[+,+,+,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,+,+,+] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[+,-,+,+] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[+,+,-,+] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[+,+,+,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,+,+] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[-,+,-,+] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[-,+,+,-] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[+,-,-,+] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[+,-,+,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[+,+,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,-,+] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[-,-,+,-] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[-,+,-,-] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[+,-,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[+,+,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[-,+,4,3] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[+,-,4,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[-,-,4,3] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[+,3,2,+] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[-,3,2,+] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[+,3,2,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[-,3,2,-] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[+,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[-,3,4,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[+,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[-,4,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[+,4,+,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000253
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => {{1}}
=> ? ∊ {0,0}
[-] => [1] => {{1}}
=> ? ∊ {0,0}
[+,+] => [1,2] => {{1},{2}}
=> 0
[-,+] => [2,1] => {{1,2}}
=> 1
[+,-] => [1,2] => {{1},{2}}
=> 0
[-,-] => [1,2] => {{1},{2}}
=> 0
[2,1] => [2,1] => {{1,2}}
=> 1
[+,+,+] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,+,+] => [2,3,1] => {{1,2,3}}
=> 1
[+,-,+] => [1,3,2] => {{1},{2,3}}
=> 1
[+,+,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,-,+] => [3,1,2] => {{1,2,3}}
=> 1
[-,+,-] => [2,1,3] => {{1,2},{3}}
=> 1
[+,-,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,-,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[+,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[-,3,2] => [3,1,2] => {{1,2,3}}
=> 1
[2,1,+] => [2,3,1] => {{1,2,3}}
=> 1
[2,1,-] => [2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 1
[3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[3,+,1] => [2,3,1] => {{1,2,3}}
=> 1
[3,-,1] => [3,1,2] => {{1,2,3}}
=> 1
[+,+,+,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,+,+,+] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[+,-,+,+] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[+,+,-,+] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[+,+,+,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,+,+] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[-,+,-,+] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[-,+,+,-] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[+,-,-,+] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[+,-,+,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[+,+,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,-,+] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[-,-,+,-] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[-,+,-,-] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[+,-,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[+,+,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[-,+,4,3] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[+,-,4,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[-,-,4,3] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[+,3,2,+] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[-,3,2,+] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[+,3,2,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[-,3,2,-] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[+,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[-,3,4,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[+,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[-,4,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[+,4,+,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000254
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => {{1}}
=> {{1}}
=> ? ∊ {0,0}
[-] => [1] => {{1}}
=> {{1}}
=> ? ∊ {0,0}
[+,+] => [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[-,+] => [2,1] => {{1,2}}
=> {{1,2}}
=> 1
[+,-] => [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[-,-] => [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[2,1] => [2,1] => {{1,2}}
=> {{1,2}}
=> 1
[+,+,+] => [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[-,+,+] => [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[+,-,+] => [1,3,2] => {{1},{2,3}}
=> {{1},{2,3}}
=> 1
[+,+,-] => [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[-,-,+] => [3,1,2] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[-,+,-] => [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[+,-,-] => [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},{2,3}}
=> {{1},{2,3}}
=> 1
[-,3,2] => [3,1,2] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[2,1,+] => [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[2,1,-] => [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[2,3,1] => [3,1,2] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[3,1,2] => [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[3,+,1] => [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[3,-,1] => [3,1,2] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[+,+,+,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[-,+,+,+] => [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[+,-,+,+] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[+,+,-,+] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 1
[+,+,+,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[-,-,+,+] => [3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 2
[-,+,-,+] => [2,4,1,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[-,+,+,-] => [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[+,-,-,+] => [1,4,2,3] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[+,-,+,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
[+,+,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[-,-,-,+] => [4,1,2,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[-,-,+,-] => [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[-,+,-,-] => [2,1,3,4] => {{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
[+,+,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 1
[-,+,4,3] => [2,4,1,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[+,-,4,3] => [1,4,2,3] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[-,-,4,3] => [4,1,2,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[+,3,2,+] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[-,3,2,+] => [3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 2
[+,3,2,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
[-,3,2,-] => [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[+,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[-,3,4,2] => [4,1,2,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[+,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[-,4,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 2
[+,4,+,2] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
Description
The nesting number of a set partition.
This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St001418
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1,0]
=> ? ∊ {0,0}
[-] => [1] => [1] => [1,0]
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[-,+] => [2,1] => [2,1] => [1,1,0,0]
=> 1
[+,-] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[-,-] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[-,+,+] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[+,-,+] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[-,-,+] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[-,+,-] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[+,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[-,3,2] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,1,+] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,1,-] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,+,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,-,1] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[-,+,4,3] => [2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[+,-,4,3] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[-,-,4,3] => [4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[+,3,2,+] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[-,3,2,-] => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[+,3,4,2] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[-,3,4,2] => [4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[+,4,2,3] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[-,4,2,3] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[+,4,+,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St000260
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => ([],1)
=> 0
[-] => [1] => [1] => ([],1)
=> 0
[+,+] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,0,1}
[-,+] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,0,1}
[+,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,0,1}
[-,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,0,1}
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[+,+,+] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[-,+,+] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[+,-,+] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[+,+,-] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[-,-,+] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[-,+,-] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[+,-,-] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[-,-,-] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[+,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[-,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,+] => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[2,1,-] => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [2,3,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0}
[3,1,2] => [3,1,2] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0}
[3,+,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0}
[3,-,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0}
[+,+,+,+] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,+,+,+] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,-,+,+] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,+,-,+] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,+,+,-] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,-,+,+] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,+,-,+] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,+,+,-] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,-,-,+] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,-,+,-] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,+,-,-] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,-,-,+] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,-,+,-] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,+,-,-] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,-,-,-] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,-,-,-] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,+,4,3] => [1,2,4,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,+,4,3] => [1,2,4,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,-,4,3] => [1,2,4,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,-,4,3] => [1,2,4,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,3,2,+] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[-,3,2,+] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[+,3,2,-] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[-,3,2,-] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[+,3,4,2] => [1,3,4,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[-,3,4,2] => [1,3,4,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,4,2,3] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[-,4,2,3] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[+,4,+,2] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[-,4,+,2] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[+,4,-,2] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[-,4,-,2] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,+,+] => [2,1,3,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,-,+] => [2,1,3,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,+,-] => [2,1,3,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,-,-] => [2,1,3,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1,+] => [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,-] => [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,4,1] => [2,3,4,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [2,4,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,+,1] => [2,4,3,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,-,1] => [2,4,3,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2] => [3,1,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,+,1,+] => [3,2,1,4] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,-,1,+] => [3,2,1,4] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,+,1,-] => [3,2,1,4] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,-,1,-] => [3,2,1,4] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,+,4,1] => [3,2,4,1] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,-,4,1] => [3,2,4,1] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,1,2] => [3,4,1,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,+,1,3] => [4,2,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,-,1,3] => [4,2,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [4,3,1,2] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [4,3,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[+,3,5,+,2] => [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,3,5,+,2] => [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,3,5,-,2] => [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,3,5,-,2] => [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,4,+,2,-] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,4,-,2,+] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,4,+,2,-] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,4,-,2,-] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,4,-,2,-] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,4,+,5,2] => [1,4,3,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,4,+,5,2] => [1,4,3,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,4,-,5,2] => [1,4,3,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,4,-,5,2] => [1,4,3,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,4,5,3,2] => [1,4,5,3,2] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,4,5,3,2] => [1,4,5,3,2] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,5,+,+,2] => [1,5,3,4,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[-,5,+,+,2] => [1,5,3,4,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[+,5,-,+,2] => [1,5,3,4,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000259The diameter of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001864The number of excedances of a signed permutation. St000456The monochromatic index of a connected graph. St001823The Stasinski-Voll length of a signed permutation.
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