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Your data matches 71 different statistics following compositions of up to 3 maps.
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Matching statistic: St001397
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Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> 0
[1,0,1,0]
=> ([(0,1)],2)
=> 0
[1,1,0,0]
=> ([(0,1)],2)
=> 0
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
Description
Number of pairs of incomparable elements in a finite poset.
For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
Matching statistic: St000987
Values
[1,0]
=> ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001341
Values
[1,0]
=> ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
Description
The number of edges in the center of a graph.
The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
Matching statistic: St001479
Values
[1,0]
=> ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
Description
The number of bridges of a graph.
A bridge is an edge whose removal increases the number of connected components of the graph.
Matching statistic: St001512
Values
[1,0]
=> ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
Description
The minimum rank of a graph.
The minimum rank of a simple graph G is the smallest possible rank over all symmetric real matrices whose entry in row $i$ and column $j$ (for $i\neq j$) is nonzero whenever $\{i, j\}$ is an edge in
$G$, and zero otherwise.
Matching statistic: St001736
Values
[1,0]
=> ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[1,1,0,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
Description
The total number of cycles in a graph.
For example, the complete graph on four vertices has four triangles and three different four-cycles.
Matching statistic: St000123
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
* The Simion-Schmidt map takes a permutation and turns each occurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
* It is the number of pairs of positions for the pattern letters 2 and 1 in occurrences of 321 in a permutation. Thus, for a permutation $\pi$ this is the number of pairs $(j,k)$ such that there exists an index $i$ satisfying $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. See also [[St000119]] and [[St000371]].
* Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Matching statistic: St000220
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => [2,3,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => [3,1,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,3,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => [5,3,4,2,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [5,2,3,1,4] => [5,2,4,1,3] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => [4,5,2,1,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,5,1,2,3] => [4,5,1,3,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => [4,3,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => [4,3,5,1,2] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [4,3,2,5,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,3,1,2,5] => [4,3,1,5,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,5,1] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,1,5] => [4,2,5,1,3] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,5,4,2,1] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,4,5,1,2] => [3,5,4,1,2] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => [3,5,2,4,1] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => [3,5,2,1,4] => 1
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St000357
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000357: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000357: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1
Description
The number of occurrences of the pattern 12-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $12\!\!-\!\!3$.
Matching statistic: St000360
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000360: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000360: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
Description
The number of occurrences of the pattern 32-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $32\!\!-\!\!1$.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St001311The cyclomatic number of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000081The number of edges of a graph. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001649The length of a longest trail in a graph. St001869The maximum cut size of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000454The largest eigenvalue of a graph if it is integral. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. 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. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St000119The number of occurrences of the pattern 321 in a permutation. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000647The number of big descents of a permutation. St000731The number of double exceedences of a permutation. St001394The genus of a permutation. St000007The number of saliances of the permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St000842The breadth of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000534The number of 2-rises of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000451The length of the longest pattern of the form k 1 2. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001153The number of blocks with even minimum in a set partition. St000028The number of stack-sorts needed to sort a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation.
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