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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St001632
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,4),(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001960
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ? = 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ? = 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => ? = 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ? = 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => ? = 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ? = 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? = 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? = 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ? = 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [6,8,1,5,2,7,3,4] => ? = 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [6,3,1,2,4,7,8,5] => ? = 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? = 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000664
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Values
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? = 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ? = 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ? = 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? = 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? = 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? = 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? = 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => ? = 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ? = 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => ? = 1
Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St001207
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Values
[2]
=> []
=> []
=> [1] => ? = 0 + 2
[1,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 0 + 2
[3]
=> []
=> []
=> [1] => ? = 1 + 2
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[4]
=> []
=> []
=> [1] => ? = 1 + 2
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 0 + 2
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 0 + 2
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 0 + 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 2
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 0 + 2
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 0 + 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 0 + 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 0 + 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? = 0 + 2
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 0 + 2
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? = 0 + 2
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 2
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 2
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001526
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Values
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 1 + 3
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 3
[2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 3
[1,1,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 0 + 3
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 0 + 3
[2,2,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 3
[4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 3
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 3
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[2,2,2]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 3
[2,2,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 3
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 3
[4,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 3
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 3
[3,2,2]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 1 + 3
[4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 3
[3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 0 + 3
[3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 0 + 3
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 3
[4,3,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 0 + 3
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1 + 3
[3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 3
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001556
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Values
[2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[3]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[4]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[3,2]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0
[3,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0
[2,2,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
[4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 0
[4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[3,3]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1
[3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 0
[2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 0
[2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 2
[4,3]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 0
[3,3,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 1
[3,2,2]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => ? = 0
[2,2,2,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? = 1
[4,2,2]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ? = 0
[3,3,2]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => ? = 0
[3,3,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => ? = 0
[4,3,2]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [8,1,2,3,6,7,4,5] => ? = 0
[4,3,1,1]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,1,2,3,9,8,4,6,7] => ? = 0
[4,2,2,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,1,2,3,4,5,8,9,6] => ? = 1
[3,3,2,1]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001686
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Values
[2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1 = 0 + 1
[1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1 = 0 + 1
[3]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 2 = 1 + 1
[1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 1 = 0 + 1
[4]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[2,2]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1 = 0 + 1
[3,2]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 0 + 1
[3,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 + 1
[2,2,1]
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[4,2]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 0 + 1
[4,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 0 + 1
[3,3]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? = 1 + 1
[3,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 + 1
[2,2,2]
=> [6]
=> [[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 0 + 1
[2,2,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 2 + 1
[4,3]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[4,1,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 + 1
[3,3,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 1 + 1
[3,2,2]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 0 + 1
[2,2,2,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[4,2,2]
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 0 + 1
[3,3,2]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [[5,2,1,0,0,0,0,0],[4,2,1,0,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? = 0 + 1
[3,3,1,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [[3,2,1,1,1,0,0,0],[2,2,1,1,1,0,0],[2,1,1,1,1,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 0 + 1
[4,3,2]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 0 + 1
[4,3,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [[5,2,1,1,0,0,0,0,0],[4,2,1,1,0,0,0,0],[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 0 + 1
[4,2,2,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[3,3,2,1]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[3,3,3,0,0,0,0,0,0],[3,3,2,0,0,0,0,0],[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
Description
The order of promotion on a Gelfand-Tsetlin pattern.
Matching statistic: St000937
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Values
[2]
=> []
=> []
=> ?
=> ? = 0
[1,1]
=> [1]
=> [1]
=> []
=> ? = 0
[3]
=> []
=> []
=> ?
=> ? = 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0
[4]
=> []
=> []
=> ?
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0
[3,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [3]
=> []
=> ? = 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0
[2,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> [2]
=> 2
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[4,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0
[3,3,2]
=> [3,2]
=> [5]
=> []
=> ? = 0
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> [1]
=> ? = 0
[4,3,2]
=> [3,2]
=> [5]
=> []
=> ? = 0
[4,3,1,1]
=> [3,1,1]
=> [4,1]
=> [1]
=> ? = 0
[4,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[3,3,2,1]
=> [3,2,1]
=> [5,1]
=> [1]
=> ? = 1
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000478
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Values
[2]
=> []
=> []
=> ?
=> ? = 0 - 1
[1,1]
=> [1]
=> [1]
=> []
=> ? = 0 - 1
[3]
=> []
=> []
=> ?
=> ? = 1 - 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0 - 1
[4]
=> []
=> []
=> ?
=> ? = 1 - 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0 - 1
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0 - 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0 - 1
[2,2,1]
=> [2,1]
=> [3]
=> []
=> ? = 1 - 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0 - 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0 - 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[2,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0 - 1
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0 - 1
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[4,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0 - 1
[3,3,2]
=> [3,2]
=> [5]
=> []
=> ? = 0 - 1
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> [1]
=> ? = 0 - 1
[4,3,2]
=> [3,2]
=> [5]
=> []
=> ? = 0 - 1
[4,3,1,1]
=> [3,1,1]
=> [4,1]
=> [1]
=> ? = 0 - 1
[4,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[3,3,2,1]
=> [3,2,1]
=> [5,1]
=> [1]
=> ? = 1 - 1
Description
Another weight of a partition according to Alladi.
According to Theorem 3.4 (Alladi 2012) in [1]
$$
\sum_{\pi\in GG_1(r)} w_1(\pi)
$$
equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
Matching statistic: St000934
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Values
[2]
=> []
=> []
=> ?
=> ? = 0 - 1
[1,1]
=> [1]
=> [1]
=> []
=> ? = 0 - 1
[3]
=> []
=> []
=> ?
=> ? = 1 - 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0 - 1
[4]
=> []
=> []
=> ?
=> ? = 1 - 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0 - 1
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0 - 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0 - 1
[2,2,1]
=> [2,1]
=> [3]
=> []
=> ? = 1 - 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> ? = 0 - 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 0 - 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[2,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0 - 1
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0 - 1
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[4,2,2]
=> [2,2]
=> [4]
=> []
=> ? = 0 - 1
[3,3,2]
=> [3,2]
=> [5]
=> []
=> ? = 0 - 1
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> [1]
=> ? = 0 - 1
[4,3,2]
=> [3,2]
=> [5]
=> []
=> ? = 0 - 1
[4,3,1,1]
=> [3,1,1]
=> [4,1]
=> [1]
=> ? = 0 - 1
[4,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[3,3,2,1]
=> [3,2,1]
=> [5,1]
=> [1]
=> ? = 1 - 1
Description
The 2-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000455The second largest eigenvalue of a graph if it is integral. St001857The number of edges in the reduced word graph of a signed permutation. St000260The radius of a connected graph. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001344The neighbouring number of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001722The number of minimal chains with small intervals between a binary word and the top element. St000075The orbit size of a standard tableau under promotion. St000259The diameter of a connected graph. St000527The width of the poset. St001517The length of a longest pair of twins in a permutation. St000068The number of minimal elements in a poset. St001060The distinguishing index of a graph. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000022The number of fixed points of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000454The largest eigenvalue of a graph if it is integral. St000871The number of very big ascents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000662The staircase size of the code of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000842The breadth of a permutation. St000862The number of parts of the shifted shape of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000891The number of distinct diagonal sums of a permutation matrix. St001090The number of pop-stack-sorts needed to sort a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.
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