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Your data matches 160 different statistics following compositions of up to 3 maps.
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Matching statistic: St001479
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 0
([],4)
=> 0
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 3
([(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([],5)
=> 0
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([],6)
=> 0
([(4,5)],6)
=> 1
([(3,5),(4,5)],6)
=> 2
([(2,5),(3,5),(4,5)],6)
=> 3
([(2,5),(3,4)],6)
=> 2
([(2,5),(3,4),(4,5)],6)
=> 3
([(1,2),(3,5),(4,5)],6)
=> 3
([(3,4),(3,5),(4,5)],6)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
([],7)
=> 0
([(5,6)],7)
=> 1
([(4,6),(5,6)],7)
=> 2
([(3,6),(4,6),(5,6)],7)
=> 3
([(3,6),(4,5)],7)
=> 2
([(3,6),(4,5),(5,6)],7)
=> 3
([(2,3),(4,6),(5,6)],7)
=> 3
([(4,5),(4,6),(5,6)],7)
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
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: St000475
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 1
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([],6)
=> []
=> 0
([(4,5)],6)
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 0
([],7)
=> []
=> 0
([(5,6)],7)
=> [1]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 3
([(3,6),(4,5)],7)
=> [1,1]
=> 2
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> 3
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> 3
([(4,5),(4,6),(5,6)],7)
=> [3]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001615
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
([],4)
=> ([],1)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
([(0,2),(0,3),(1,2),(1,3)],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),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ([],1)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ([],1)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ([],1)
=> 0
([],5)
=> ([],1)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(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),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ([],1)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ([],1)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ([],1)
=> 0
([],6)
=> ([],1)
=> ([],1)
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(1,2),(3,5),(4,5)],6)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ([],1)
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ([],1)
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ([],1)
=> 0
([],7)
=> ([],1)
=> ([],1)
=> 0
([(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,5),(5,6)],7)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(2,3),(4,6),(5,6)],7)
=> ([(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),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(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),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ([],1)
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ([],1)
=> 0
Description
The number of join prime elements of a lattice.
An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001657
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00117: Graphs —Ore closure⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 0
([(4,5)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [2,2,2,1,1]
=> 3
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> [2,2,1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [2,2,2,1,1]
=> 3
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> [2,2,2,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0
([],7)
=> ([],7)
=> [1,1,1,1,1,1,1]
=> 0
([(5,6)],7)
=> ([(5,6)],7)
=> [2,1,1,1,1,1]
=> 1
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> [2,2,1,1,1,1]
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [2,2,2,1,1,1]
=> 3
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> 2
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [2,2,2,1,1,1]
=> 3
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> [2,2,2,1,1]
=> 3
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000445
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> []
=> 0
([],2)
=> []
=> []
=> []
=> 0
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([],3)
=> []
=> []
=> []
=> 0
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([],4)
=> []
=> []
=> []
=> 0
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([],5)
=> []
=> []
=> []
=> 0
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([],6)
=> []
=> []
=> []
=> 0
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([],7)
=> []
=> []
=> []
=> 0
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St001250
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00117: Graphs —Ore closure⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([],2)
=> []
=> ? = 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 1
([],3)
=> ([],3)
=> []
=> ? = 0
([(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([],4)
=> ([],4)
=> []
=> ? = 0
([(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([],5)
=> ([],5)
=> []
=> ? = 0
([(3,4)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([],6)
=> ([],6)
=> []
=> ? = 0
([(4,5)],6)
=> ([(4,5)],6)
=> [1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [1,1]
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> [1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 3
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 0
([],7)
=> ([],7)
=> []
=> ? = 0
([(5,6)],7)
=> ([(5,6)],7)
=> [1]
=> 1
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> [1,1]
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 3
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> [1,1]
=> 2
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> 3
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> 3
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> [3]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> 0
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> 3
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> 0
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St001606
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> ? = 0
([],2)
=> []
=> []
=> ? = 0
([(0,1)],2)
=> [1]
=> [1]
=> 1
([],3)
=> []
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([],4)
=> []
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([],5)
=> []
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([],6)
=> []
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([],7)
=> []
=> []
=> ? = 0
([(5,6)],7)
=> [1]
=> [1]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> [2]
=> 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> 3
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> 2
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [3]
=> 3
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> 3
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,1,1,1]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [3]
=> 3
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.
Matching statistic: St000297
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> => ? => ? = 0
([],2)
=> []
=> => ? => ? = 0
([(0,1)],2)
=> [1]
=> 10 => 10 => 1
([],3)
=> []
=> => ? => ? = 0
([(1,2)],3)
=> [1]
=> 10 => 10 => 1
([(0,2),(1,2)],3)
=> [1,1]
=> 110 => 110 => 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 0010 => 0
([],4)
=> []
=> => ? => ? = 0
([(2,3)],4)
=> [1]
=> 10 => 10 => 1
([(1,3),(2,3)],4)
=> [1,1]
=> 110 => 110 => 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 1110 => 1110 => 3
([(0,3),(1,2)],4)
=> [1,1]
=> 110 => 110 => 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 1110 => 1110 => 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1000 => 0010 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 00010 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 100000 => 000010 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1000000 => 0000010 => 0
([],5)
=> []
=> => ? => ? = 0
([(3,4)],5)
=> [1]
=> 10 => 10 => 1
([(2,4),(3,4)],5)
=> [1,1]
=> 110 => 110 => 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 1110 => 1110 => 3
([(1,4),(2,3)],5)
=> [1,1]
=> 110 => 110 => 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 1110 => 1110 => 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 1110 => 1110 => 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1000 => 0010 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 10000 => 00010 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 000010 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 0000010 => 0
([],6)
=> []
=> => ? => ? = 0
([(4,5)],6)
=> [1]
=> 10 => 10 => 1
([(3,5),(4,5)],6)
=> [1,1]
=> 110 => 110 => 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 1110 => 1110 => 3
([(2,5),(3,4)],6)
=> [1,1]
=> 110 => 110 => 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 1110 => 1110 => 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 1110 => 1110 => 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 1000 => 0010 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> 10000 => 00010 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 100000 => 000010 => 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 1110 => 1110 => 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 1000000 => 0000010 => 0
([],7)
=> []
=> => ? => ? = 0
([(5,6)],7)
=> [1]
=> 10 => 10 => 1
([(4,6),(5,6)],7)
=> [1,1]
=> 110 => 110 => 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 1110 => 1110 => 3
([(3,6),(4,5)],7)
=> [1,1]
=> 110 => 110 => 2
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> 1110 => 1110 => 3
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> 1110 => 1110 => 3
([(4,5),(4,6),(5,6)],7)
=> [3]
=> 1000 => 0010 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> 10000 => 00010 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> 100000 => 000010 => 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> 1110 => 1110 => 3
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> 1000000 => 0000010 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000968
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000968: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000968: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> []
=> ? = 0 + 1
([],2)
=> []
=> []
=> []
=> ? = 0 + 1
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([],3)
=> []
=> []
=> []
=> ? = 0 + 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
([],4)
=> []
=> []
=> []
=> ? = 0 + 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
([],5)
=> []
=> []
=> []
=> ? = 0 + 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
([],6)
=> []
=> []
=> []
=> ? = 0 + 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
([],7)
=> []
=> []
=> []
=> ? = 0 + 1
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
Description
We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. Then we calculate the dominant dimension of that CNakayama algebra.
Matching statistic: St001523
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001523: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001523: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> []
=> ? = 0 + 1
([],2)
=> []
=> []
=> []
=> ? = 0 + 1
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([],3)
=> []
=> []
=> []
=> ? = 0 + 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([],4)
=> []
=> []
=> []
=> ? = 0 + 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([],5)
=> []
=> []
=> []
=> ? = 0 + 1
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([],6)
=> []
=> []
=> []
=> ? = 0 + 1
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([],7)
=> []
=> []
=> []
=> ? = 0 + 1
([(5,6)],7)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
Description
The degree of symmetry of a Dyck path.
Given a Dyck path $D=(d_1,\dots,d_{2n})$, with $d_i\in\{0,1\}$, this is the number of positions $1\leq i\leq n$ such that $d_i = 1-d_{2n+1-i}$ and the initial height of the $i$-th step $\sum_{j < i} d_i$ equals the final height of the $(2n+1-i)$-th step.
The following 150 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000674The number of hills of a Dyck path. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000454The largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001389The number of partitions of the same length below the given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St000264The girth of a graph, which is not a tree. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001525The number of symmetric hooks on the diagonal of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000781The number of proper colouring schemes of a Ferrers diagram. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001568The smallest positive integer that does not appear twice in the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000699The toughness times the least common multiple of 1,. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000455The second largest eigenvalue of a graph if it is integral. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001060The distinguishing index of a graph. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000379The number of Hamiltonian cycles in a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001330The hat guessing number of a graph. 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. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001651The Frankl number of a lattice. St001545The second Elser number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.
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