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Your data matches 130 different statistics following compositions of up to 3 maps.
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Matching statistic: St000217
St000217: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 2
[3,4,2,1] => 0
[4,1,2,3] => 3
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 2
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 2
Description
The number of occurrences of the pattern 312 in a permutation.
Matching statistic: St000218
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 2
[3,1,4,2] => 1
[3,2,1,4] => 3
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 1
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 3
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000220
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 0
[2,1,4,3] => 2
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 4
[1,2,5,3,4] => 4
[1,2,5,4,3] => 6
[1,3,2,4,5] => 1
[1,3,2,5,4] => 4
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 4
[1,3,5,4,2] => 5
[1,4,2,3,5] => 2
[1,4,2,5,3] => 4
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 4
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St001398
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001398: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> 0
[2,1] => ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => ([(0,1),(0,2)],3)
=> 1
[2,1,3] => ([(0,2),(1,2)],3)
=> 0
[2,3,1] => ([(1,2)],3)
=> 0
[3,1,2] => ([(1,2)],3)
=> 0
[3,2,1] => ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 0
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[3,4,1,2] => ([(0,3),(1,2)],4)
=> 0
[3,4,2,1] => ([(2,3)],4)
=> 0
[4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[4,2,1,3] => ([(1,3),(2,3)],4)
=> 0
[4,2,3,1] => ([(2,3)],4)
=> 0
[4,3,1,2] => ([(2,3)],4)
=> 0
[4,3,2,1] => ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 3
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 4
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 4
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 6
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 4
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 5
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 4
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 4
Description
Number of subsets of size 3 of elements in a poset that form a "v".
For a finite poset $(P,\leq)$, this is the number of sets $\{x,y,z\} \in \binom{P}{3}$ that form a "v"-subposet (i.e., a subposet consisting of a bottom element covered by two incomparable elements).
Matching statistic: St001898
Mp00305: Permutations —parking function⟶ Parking functions
Mp00290: Parking functions —to ordered set partition⟶ Ordered set partitions
St001898: Ordered set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00290: Parking functions —to ordered set partition⟶ Ordered set partitions
St001898: Ordered set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [{1}] => 0
[1,2] => [1,2] => [{1},{2}] => 0
[2,1] => [2,1] => [{2},{1}] => 0
[1,2,3] => [1,2,3] => [{1},{2},{3}] => 0
[1,3,2] => [1,3,2] => [{1},{3},{2}] => 1
[2,1,3] => [2,1,3] => [{2},{1},{3}] => 0
[2,3,1] => [2,3,1] => [{3},{1},{2}] => 0
[3,1,2] => [3,1,2] => [{2},{3},{1}] => 0
[3,2,1] => [3,2,1] => [{3},{2},{1}] => 0
[1,2,3,4] => [1,2,3,4] => [{1},{2},{3},{4}] => 0
[1,2,4,3] => [1,2,4,3] => [{1},{2},{4},{3}] => 2
[1,3,2,4] => [1,3,2,4] => [{1},{3},{2},{4}] => 1
[1,3,4,2] => [1,3,4,2] => [{1},{4},{2},{3}] => 2
[1,4,2,3] => [1,4,2,3] => [{1},{3},{4},{2}] => 2
[1,4,3,2] => [1,4,3,2] => [{1},{4},{3},{2}] => 3
[2,1,3,4] => [2,1,3,4] => [{2},{1},{3},{4}] => 0
[2,1,4,3] => [2,1,4,3] => [{2},{1},{4},{3}] => 2
[2,3,1,4] => [2,3,1,4] => [{3},{1},{2},{4}] => 0
[2,3,4,1] => [2,3,4,1] => [{4},{1},{2},{3}] => 0
[2,4,1,3] => [2,4,1,3] => [{3},{1},{4},{2}] => 1
[2,4,3,1] => [2,4,3,1] => [{4},{1},{3},{2}] => 1
[3,1,2,4] => [3,1,2,4] => [{2},{3},{1},{4}] => 0
[3,1,4,2] => [3,1,4,2] => [{2},{4},{1},{3}] => 1
[3,2,1,4] => [3,2,1,4] => [{3},{2},{1},{4}] => 0
[3,2,4,1] => [3,2,4,1] => [{4},{2},{1},{3}] => 0
[3,4,1,2] => [3,4,1,2] => [{3},{4},{1},{2}] => 0
[3,4,2,1] => [3,4,2,1] => [{4},{3},{1},{2}] => 0
[4,1,2,3] => [4,1,2,3] => [{2},{3},{4},{1}] => 0
[4,1,3,2] => [4,1,3,2] => [{2},{4},{3},{1}] => 1
[4,2,1,3] => [4,2,1,3] => [{3},{2},{4},{1}] => 0
[4,2,3,1] => [4,2,3,1] => [{4},{2},{3},{1}] => 0
[4,3,1,2] => [4,3,1,2] => [{3},{4},{2},{1}] => 0
[4,3,2,1] => [4,3,2,1] => [{4},{3},{2},{1}] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [{1},{2},{3},{4},{5}] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [{1},{2},{3},{5},{4}] => 3
[1,2,4,3,5] => [1,2,4,3,5] => [{1},{2},{4},{3},{5}] => 2
[1,2,4,5,3] => [1,2,4,5,3] => [{1},{2},{5},{3},{4}] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [{1},{2},{4},{5},{3}] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [{1},{2},{5},{4},{3}] => 6
[1,3,2,4,5] => [1,3,2,4,5] => [{1},{3},{2},{4},{5}] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [{1},{3},{2},{5},{4}] => 4
[1,3,4,2,5] => [1,3,4,2,5] => [{1},{4},{2},{3},{5}] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [{1},{5},{2},{3},{4}] => 3
[1,3,5,2,4] => [1,3,5,2,4] => [{1},{4},{2},{5},{3}] => 4
[1,3,5,4,2] => [1,3,5,4,2] => [{1},{5},{2},{4},{3}] => 5
[1,4,2,3,5] => [1,4,2,3,5] => [{1},{3},{4},{2},{5}] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [{1},{3},{5},{2},{4}] => 4
[1,4,3,2,5] => [1,4,3,2,5] => [{1},{4},{3},{2},{5}] => 3
[1,4,3,5,2] => [1,4,3,5,2] => [{1},{5},{3},{2},{4}] => 4
[1,4,5,2,3] => [1,4,5,2,3] => [{1},{4},{5},{2},{3}] => 4
Description
The number of occurrences of an 132 pattern in an ordered set partition.
An occurrence of a pattern $\pi\in\mathfrak S_k$ ordered set partition with blocks $B_1|\dots|B_\ell$ is a sequence of elements $e_1,\dots,e_k$ with $e_i\in B_{j_i}$ and $j_1 < \dots < j_k$ order-isomorphic to $\pi$.
Matching statistic: St000219
St000219: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => ? = 0
[1,2] => ? ∊ {0,0}
[2,1] => ? ∊ {0,0}
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 3
[2,4,1,3] => 1
[2,4,3,1] => 2
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 1
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 3
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
[1,5,2,3,4] => 0
[1,5,2,4,3] => 0
Description
The number of occurrences of the pattern 231 in a permutation.
Matching statistic: St000376
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 78%●distinct values known / distinct values provided: 71%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 78%●distinct values known / distinct values provided: 71%
Values
[1] => [1,0]
=> []
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> []
=> []
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,3}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,3}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,3}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,3}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,3}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,3}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
Description
The bounce deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$$
The zeta map [[Mp00032]] sends this statistic to the dinv deficit [[St000369]], both are thus equidistributed.
Matching statistic: St001438
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 71% ●values known / values provided: 75%●distinct values known / distinct values provided: 71%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 71% ●values known / values provided: 75%●distinct values known / distinct values provided: 71%
Values
[1] => [1,0]
=> [1,0]
=> [[1],[]]
=> 0
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {0,3}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {0,3}
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
Description
The number of missing boxes of a skew partition.
Matching statistic: St001502
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001502: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001502: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> []
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1,2,2}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1,2,2}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 4
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> 6
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 4
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6}
Description
The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras.
We use the code below to translate them to Dyck paths.
The algebras where the statistic returns 0 are exactly the higher Auslander algebras and are of special interest. It seems like they are counted by the number of divisors function.
Matching statistic: St001651
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ? = 0
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[2,1] => ([],2)
=> ([],1)
=> ? = 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {0,0,0}
[2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,0,0}
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> 0
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 0
[3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0}
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 0
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 0
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 0
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 0
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,2,2,2,3}
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 3
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 0
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 0
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 0
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 0
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 0
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 0
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 0
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 0
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 0
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 0
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 0
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 0
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 0
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 0
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6}
Description
The Frankl number of a lattice.
For a lattice $L$ on at least two elements, this is
$$
\max_x(|L|-2|[x, 1]|),
$$
where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.
The following 120 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000940The number of characters of the symmetric group whose value on the partition is zero. St001279The sum of the parts of an integer partition that are at least two. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. 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. St001175The size of a partition minus the hook length of the base cell. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000929The constant term of the character polynomial of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000319The spin of an integer partition. St000320The dinv adjustment of 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. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000422The energy of a graph, if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001845The number of join irreducibles minus the rank of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001846The number of elements which do not have a complement in the lattice. St000456The monochromatic index of a connected graph. St000527The width of the poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000699The toughness times the least common multiple of 1,. 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. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001902The number of potential covers of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St001330The hat guessing number of a graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. 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. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. 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. St000706The product of the factorials of the multiplicities 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. 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. 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. St001568The smallest positive integer that does not appear twice in the partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001545The second Elser number of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000264The girth of a graph, which is not a tree. St000914The sum of the values of the Möbius function of a poset.
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