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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St000375
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000375: 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 non weak exceedences of a permutation that are 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 exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St001513
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St001513: 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] => [3,1,2] => 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] => [4,1,2,3] => 0
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => 0
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => 0
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 0
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => 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] => [3,1,2,4] => 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] => [3,1,4,2] => 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 nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St001549
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[+,-,+] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,1,+] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,-,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 0
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 0
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[-,3,2,+] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 0
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St001845
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => ([],1)
=> ([(0,1)],2)
=> 0
[-] => [1] => ([],1)
=> ([(0,1)],2)
=> 0
[+,+] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[-,+] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[+,-] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[-,-] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[2,1] => [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[+,+,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[+,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[-,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[2,1,+] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[2,1,-] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[2,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[3,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[3,+,1] => [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[3,-,1] => [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[+,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,-,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,-,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,-,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[+,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[+,3,2,+] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[-,3,2,+] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[+,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[-,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[+,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 0
[-,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 0
[+,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 0
[2,4,+,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[2,4,-,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[3,+,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[3,-,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,1,+,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[4,1,-,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[4,+,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[4,-,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[4,+,+,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,-,+,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,+,-,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,-,-,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 0
[+,+,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[-,+,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[+,-,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[+,+,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[-,-,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[-,+,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[+,-,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[-,-,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[+,3,5,+,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[-,3,5,+,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[+,3,5,-,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[-,3,5,-,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[+,4,+,2,+] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[-,4,+,2,+] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[+,4,-,2,+] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[+,4,+,2,-] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[-,4,-,2,+] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[-,4,+,2,-] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[+,4,-,2,-] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[-,4,-,2,-] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[+,4,+,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[-,4,+,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[+,4,-,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[-,4,-,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[+,4,5,2,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[-,4,5,2,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[+,4,5,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 0
[-,4,5,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 0
[+,5,2,+,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[-,5,2,+,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[+,5,2,-,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[-,5,2,-,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[+,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[-,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[+,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St000068
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
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] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,+,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [2,3,1] => ([(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)
=> 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)
=> 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)
=> 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)
=> 1 = 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)
=> 1 = 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)
=> 1 = 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)
=> 1 = 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)
=> 1 = 0 + 1
[+,3,4,2] => [1,3,4,2] => [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 = 0 + 1
[-,3,4,2] => [1,3,4,2] => [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 = 0 + 1
[+,4,2,3] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,+,4,1] => [3,2,4,1] => [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
[3,-,4,1] => [3,2,4,1] => [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
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[+,+,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,+,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,-,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,+,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,-,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,+,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,-,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,-,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,+,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,+,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,-,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,-,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,3,2,+,+] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,3,2,+,+] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,3,2,-,+] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,3,2,+,-] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,3,2,-,+] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,3,2,+,-] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,3,2,-,-] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,3,2,-,-] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[+,3,4,2,+] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[-,3,4,2,+] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[+,3,4,2,-] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[-,3,4,2,-] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[+,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[+,3,5,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,3,5,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,3,5,+,2] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[-,3,5,+,2] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[+,3,5,-,2] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[-,3,5,-,2] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[+,4,2,3,+] => [1,4,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,4,2,3,+] => [1,4,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,4,2,3,-] => [1,4,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,4,2,3,-] => [1,4,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,4,2,5,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St001719
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => ([(0,1)],2)
=> 1 = 0 + 1
[-] => [1] => ([(0,1)],2)
=> 1 = 0 + 1
[+,+] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,+] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,-] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,-] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,+,+] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[-,+,+] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[+,-,+] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[-,-,+] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-,+,-] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-,3,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,1,+] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,3,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,1,2] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,+,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,-,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[+,+,+,+] => [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 + 1
[-,+,+,+] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[+,-,+,+] => [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)
=> 1 = 0 + 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)
=> 1 = 0 + 1
[+,+,+,-] => [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 + 1
[-,-,+,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
[-,+,-,+] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[-,+,+,-] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[+,-,-,+] => [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)
=> 1 = 0 + 1
[+,-,+,-] => [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)
=> 1 = 0 + 1
[+,+,-,-] => [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 + 1
[-,-,-,+] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[-,-,+,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[-,+,-,-] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 0 + 1
[+,-,-,-] => [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 + 1
[-,-,-,-] => [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 + 1
[+,+,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 0 + 1
[-,+,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 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)
=> 1 = 0 + 1
[-,-,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)
=> 1 = 0 + 1
[+,3,2,+] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[-,3,2,+] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
[+,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[-,3,2,-] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[+,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[-,3,4,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[+,4,2,3] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 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)
=> 1 = 0 + 1
[+,4,+,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[-,4,+,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
[+,4,-,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[-,4,-,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[+,+,+,+,+] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[-,+,+,+,+] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[+,-,+,+,+] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[+,+,-,+,+] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,+,+,-,+] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 0 + 1
[+,+,+,+,-] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[-,-,+,+,+] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[-,+,+,+,-] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[+,-,-,+,+] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[+,-,+,+,-] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[+,+,-,-,+] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,+,-,+,-] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[+,+,+,-,-] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[-,-,-,+,+] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[-,-,+,+,-] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[-,+,+,-,-] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,-,-,-,+] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[+,-,-,+,-] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[+,-,+,-,-] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[+,+,-,-,-] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[-,-,-,-,+] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[-,-,-,+,-] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[-,-,+,-,-] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[-,+,-,-,-] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 0 + 1
[+,-,-,-,-] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[-,-,-,-,-] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[+,+,+,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 0 + 1
[+,+,-,5,4] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,-,-,5,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[-,-,-,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[+,+,4,3,+] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,-,4,3,+] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[+,+,4,3,-] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[-,-,4,3,+] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,-,4,3,-] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[-,-,4,3,-] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[+,+,4,5,3] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,-,4,5,3] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[-,-,4,5,3] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[+,+,5,3,4] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,-,5,3,4] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[-,-,5,3,4] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,+,5,+,3] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[+,-,5,+,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[+,+,5,-,3] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001429
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001429: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001429: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1] => [-1] => 1 = 0 + 1
[-] => [1] => [1] => [-1] => 1 = 0 + 1
[+,+] => [1,2] => [1,2] => [-2,1] => 1 = 0 + 1
[-,+] => [1,2] => [1,2] => [-2,1] => 1 = 0 + 1
[+,-] => [1,2] => [1,2] => [-2,1] => 1 = 0 + 1
[-,-] => [1,2] => [1,2] => [-2,1] => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => [1,-2] => 1 = 0 + 1
[+,+,+] => [1,2,3] => [1,2,3] => [-3,1,2] => 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,2,3] => [-3,1,2] => 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,2,3] => [-3,1,2] => 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,2,3] => [-3,1,2] => 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,2,3] => [-3,1,2] => 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,2,3] => [-3,1,2] => 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,2,3] => [-3,1,2] => 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,2,3] => [-3,1,2] => 1 = 0 + 1
[+,3,2] => [1,3,2] => [1,3,2] => [-3,2,1] => 1 = 0 + 1
[-,3,2] => [1,3,2] => [1,3,2] => [-3,2,1] => 1 = 0 + 1
[2,1,+] => [2,1,3] => [2,1,3] => [1,-3,2] => 1 = 0 + 1
[2,1,-] => [2,1,3] => [2,1,3] => [1,-3,2] => 1 = 0 + 1
[2,3,1] => [2,3,1] => [2,3,1] => [2,1,-3] => 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,1,2] => [2,-3,1] => 1 = 0 + 1
[3,+,1] => [3,2,1] => [3,2,1] => [1,2,-3] => 1 = 0 + 1
[3,-,1] => [3,2,1] => [3,2,1] => [1,2,-3] => 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [-4,1,3,2] => 1 = 0 + 1
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => [-4,1,3,2] => 1 = 0 + 1
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => [-4,1,3,2] => 1 = 0 + 1
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => [-4,1,3,2] => 1 = 0 + 1
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => [-4,2,1,3] => 1 = 0 + 1
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => [-4,2,1,3] => 1 = 0 + 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [-4,2,1,3] => 1 = 0 + 1
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => [-4,2,1,3] => 1 = 0 + 1
[+,3,4,2] => [1,3,4,2] => [1,3,4,2] => [-4,2,3,1] => 1 = 0 + 1
[-,3,4,2] => [1,3,4,2] => [1,3,4,2] => [-4,2,3,1] => 1 = 0 + 1
[+,4,2,3] => [1,4,2,3] => [1,4,2,3] => [-4,3,1,2] => 1 = 0 + 1
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0 + 1
[-,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0 + 1
[+,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0 + 1
[+,+,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0 + 1
[-,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0 + 1
[-,+,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0 + 1
[+,-,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0 + 1
[-,-,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,4,3] => ? = 0 + 1
[+,+,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0 + 1
[-,+,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0 + 1
[+,-,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0 + 1
[+,+,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0 + 1
[-,-,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0 + 1
[-,+,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0 + 1
[+,-,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0 + 1
[-,-,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [-5,1,3,2,4] => ? = 0 + 1
[+,+,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => [-5,1,3,4,2] => ? = 0 + 1
[-,+,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => [-5,1,3,4,2] => ? = 0 + 1
[+,-,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => [-5,1,3,4,2] => ? = 0 + 1
[-,-,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => [-5,1,3,4,2] => ? = 0 + 1
[+,+,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [-5,1,4,2,3] => ? = 0 + 1
[-,+,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [-5,1,4,2,3] => ? = 0 + 1
[+,-,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [-5,1,4,2,3] => ? = 0 + 1
[-,-,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [-5,1,4,2,3] => ? = 0 + 1
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0 + 1
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0 + 1
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0 + 1
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0 + 1
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0 + 1
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0 + 1
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0 + 1
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-5,1,4,3,2] => ? = 0 + 1
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-5,2,1,4,3] => ? = 0 + 1
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-5,2,1,4,3] => ? = 0 + 1
[+,3,4,2,+] => [1,3,4,2,5] => [1,3,4,2,5] => [-5,2,3,1,4] => ? = 0 + 1
[-,3,4,2,+] => [1,3,4,2,5] => [1,3,4,2,5] => [-5,2,3,1,4] => ? = 0 + 1
[+,3,4,2,-] => [1,3,4,2,5] => [1,3,4,2,5] => [-5,2,3,1,4] => ? = 0 + 1
[-,3,4,2,-] => [1,3,4,2,5] => [1,3,4,2,5] => [-5,2,3,1,4] => ? = 0 + 1
[+,3,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => [-5,2,3,4,1] => ? = 0 + 1
[-,3,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => [-5,2,3,4,1] => ? = 0 + 1
[+,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-5,2,4,3,1] => ? = 0 + 1
[-,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-5,2,4,3,1] => ? = 0 + 1
[+,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-5,2,4,3,1] => ? = 0 + 1
[-,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-5,2,4,3,1] => ? = 0 + 1
[+,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-5,3,1,4,2] => ? = 0 + 1
[-,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-5,3,1,4,2] => ? = 0 + 1
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-5,3,2,1,4] => ? = 0 + 1
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-5,3,2,1,4] => ? = 0 + 1
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-5,3,2,1,4] => ? = 0 + 1
[+,4,+,2,-] => [1,4,3,2,5] => [1,4,3,2,5] => [-5,3,2,1,4] => ? = 0 + 1
Description
The number of negative entries in a signed permutation.
Matching statistic: St001771
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 0
[+,-,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[-,+,-] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[2,1,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[3,-,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 0
[+,-,+,+] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 0
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[-,-,+,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0
[-,+,-,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+,+] => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[+,-,+,+,+] => [1,3,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[+,+,-,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[+,+,+,-,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[+,-,-,+,+] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 0
[+,-,+,-,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[-,-,+,+,-] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,-,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,+,-,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,+,+,-,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,-,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,+,+,5,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[-,-,+,5,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,-,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[+,+,4,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,+,4,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[+,-,4,3,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[-,+,4,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,4,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,5,3,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[-,-,5,3,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,5,+,3] => [2,4,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[+,+,5,-,3] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,+,5,-,3] => [2,3,1,5,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[+,-,5,-,3] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[-,-,5,-,3] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[+,3,2,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[-,3,2,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[-,3,2,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,3,2,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,2,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,2,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,4,2,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,3,4,2,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,4,5,2] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,5,2,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,5,+,2] => [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
[+,3,5,-,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,3,5,-,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[+,4,2,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,4,2,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[-,4,2,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,4,2,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[+,4,+,2,+] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[-,4,+,2,+] => [3,2,5,1,4] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 0
[+,4,-,2,+] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001870
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 0
[+,-,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[-,+,-] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[2,1,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[3,-,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 0
[+,-,+,+] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 0
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[-,-,+,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0
[-,+,-,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+,+] => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[+,-,+,+,+] => [1,3,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[+,+,-,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[+,+,+,-,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[+,-,-,+,+] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 0
[+,-,+,-,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[-,-,+,+,-] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,-,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,+,-,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,+,+,-,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,-,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,+,+,5,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[-,-,+,5,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,-,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[+,+,4,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,+,4,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[+,-,4,3,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[-,+,4,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,4,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,5,3,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[-,-,5,3,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,5,+,3] => [2,4,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[+,+,5,-,3] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,+,5,-,3] => [2,3,1,5,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[+,-,5,-,3] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[-,-,5,-,3] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[+,3,2,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[-,3,2,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[-,3,2,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,3,2,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,2,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,2,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,4,2,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,3,4,2,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,4,5,2] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,5,2,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,5,+,2] => [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
[+,3,5,-,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,3,5,-,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[+,4,2,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,4,2,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[-,4,2,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,4,2,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[+,4,+,2,+] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[-,4,+,2,+] => [3,2,5,1,4] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 0
[+,4,-,2,+] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 0
[+,-,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[-,+,-] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[2,1,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[3,-,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 0
[+,-,+,+] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 0
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[-,-,+,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0
[-,+,-,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+,+] => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[+,-,+,+,+] => [1,3,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[+,+,-,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[+,+,+,-,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[+,-,-,+,+] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 0
[+,-,+,-,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[-,-,+,+,-] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,-,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,+,-,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,+,+,-,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,-,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,+,+,5,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[-,-,+,5,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,-,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[+,+,4,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,+,4,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[+,-,4,3,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[-,+,4,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,4,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,+,5,3,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[-,-,5,3,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[-,+,5,+,3] => [2,4,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[+,+,5,-,3] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,+,5,-,3] => [2,3,1,5,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[+,-,5,-,3] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[-,-,5,-,3] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[+,3,2,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[-,3,2,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[-,3,2,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,3,2,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,2,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,2,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,4,2,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[-,3,4,2,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,4,5,2] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-,3,5,2,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[-,3,5,+,2] => [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
[+,3,5,-,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,3,5,-,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[+,4,2,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[-,4,2,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[-,4,2,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-,4,2,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[+,4,+,2,+] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[-,4,+,2,+] => [3,2,5,1,4] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 0
[+,4,-,2,+] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
Description
The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001866The nesting alignments of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001490The number of connected components of a skew partition. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001964The interval resolution global dimension of a poset. St001889The size of the connectivity set of a signed permutation. St000456The monochromatic index of a connected graph. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001768The number of reduced words of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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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