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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000209
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => []
=> []
=> [1] => 0
[-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,2] => []
=> []
=> [1] => 0
[1,-2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[2,1] => []
=> []
=> [1] => 0
[2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[-2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[-2,-1] => []
=> []
=> [1] => 0
[1,2,3] => []
=> []
=> [1] => 0
[1,2,-3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[-1,2,3] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3
[1,3,2] => []
=> []
=> [1] => 0
[1,3,-2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[1,-3,2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[1,-3,-2] => []
=> []
=> [1] => 0
[-1,3,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[-1,-3,-2] => [1]
=> [1,0]
=> [2,1] => 1
[2,1,3] => []
=> []
=> [1] => 0
[2,1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[-2,1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[-2,-1,3] => []
=> []
=> [1] => 0
[-2,-1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,3,1] => []
=> []
=> [1] => 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[2,-3,-1] => []
=> []
=> [1] => 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-2,3,-1] => []
=> []
=> [1] => 0
[-2,-3,1] => []
=> []
=> [1] => 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[3,1,2] => []
=> []
=> [1] => 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[3,-1,-2] => []
=> []
=> [1] => 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-3,1,-2] => []
=> []
=> [1] => 0
[-3,-1,2] => []
=> []
=> [1] => 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
Description
Maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St001190
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001190: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 83%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001190: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 83%
Values
[1] => []
=> []
=> []
=> ? = 0 + 1
[-1] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[1,2] => []
=> []
=> []
=> ? = 0 + 1
[1,-2] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[-1,2] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[-1,-2] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[2,1] => []
=> []
=> []
=> ? = 0 + 1
[2,-1] => [2]
=> [1,1]
=> [1,1,0,0]
=> 3 = 2 + 1
[-2,1] => [2]
=> [1,1]
=> [1,1,0,0]
=> 3 = 2 + 1
[-2,-1] => []
=> []
=> []
=> ? = 0 + 1
[1,2,3] => []
=> []
=> []
=> ? = 0 + 1
[1,2,-3] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[1,-2,3] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[1,-2,-3] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[-1,2,3] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[-1,2,-3] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[-1,-2,3] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2] => []
=> []
=> []
=> ? = 0 + 1
[1,3,-2] => [2]
=> [1,1]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,-3,2] => [2]
=> [1,1]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,-3,-2] => []
=> []
=> []
=> ? = 0 + 1
[-1,3,2] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[-1,3,-2] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[-1,-3,2] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[-1,-3,-2] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[2,1,3] => []
=> []
=> []
=> ? = 0 + 1
[2,1,-3] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[2,-1,3] => [2]
=> [1,1]
=> [1,1,0,0]
=> 3 = 2 + 1
[2,-1,-3] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[-2,1,3] => [2]
=> [1,1]
=> [1,1,0,0]
=> 3 = 2 + 1
[-2,1,-3] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[-2,-1,3] => []
=> []
=> []
=> ? = 0 + 1
[-2,-1,-3] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[2,3,1] => []
=> []
=> []
=> ? = 0 + 1
[2,3,-1] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[2,-3,1] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[2,-3,-1] => []
=> []
=> []
=> ? = 0 + 1
[-2,3,1] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[-2,3,-1] => []
=> []
=> []
=> ? = 0 + 1
[-2,-3,1] => []
=> []
=> []
=> ? = 0 + 1
[-2,-3,-1] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[3,1,2] => []
=> []
=> []
=> ? = 0 + 1
[3,1,-2] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[3,-1,2] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[3,-1,-2] => []
=> []
=> []
=> ? = 0 + 1
[-3,1,2] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[-3,1,-2] => []
=> []
=> []
=> ? = 0 + 1
[-3,-1,2] => []
=> []
=> []
=> ? = 0 + 1
[-3,-1,-2] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[3,2,1] => []
=> []
=> []
=> ? = 0 + 1
[3,2,-1] => [2]
=> [1,1]
=> [1,1,0,0]
=> 3 = 2 + 1
[3,-2,1] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[3,-2,-1] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[-3,2,1] => [2]
=> [1,1]
=> [1,1,0,0]
=> 3 = 2 + 1
[-3,2,-1] => []
=> []
=> []
=> ? = 0 + 1
[-3,-2,1] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[-3,-2,-1] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[1,2,3,4] => []
=> []
=> []
=> ? = 0 + 1
[1,2,3,-4] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[1,2,-3,4] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[1,2,-3,-4] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[1,-2,3,4] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[1,-2,3,-4] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[1,-2,-3,4] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[1,-2,-3,-4] => [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[-1,2,3,4] => [1]
=> [1]
=> [1,0]
=> 2 = 1 + 1
[-1,2,3,-4] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[-1,2,-3,4] => [1,1]
=> [2]
=> [1,0,1,0]
=> 3 = 2 + 1
[-1,2,-3,-4] => [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3] => []
=> []
=> []
=> ? = 0 + 1
[1,2,-4,-3] => []
=> []
=> []
=> ? = 0 + 1
[1,3,2,4] => []
=> []
=> []
=> ? = 0 + 1
[1,-3,-2,4] => []
=> []
=> []
=> ? = 0 + 1
[1,3,4,2] => []
=> []
=> []
=> ? = 0 + 1
[1,3,-4,-2] => []
=> []
=> []
=> ? = 0 + 1
[1,-3,4,-2] => []
=> []
=> []
=> ? = 0 + 1
[1,-3,-4,2] => []
=> []
=> []
=> ? = 0 + 1
[1,4,2,3] => []
=> []
=> []
=> ? = 0 + 1
[1,4,-2,-3] => []
=> []
=> []
=> ? = 0 + 1
[1,-4,2,-3] => []
=> []
=> []
=> ? = 0 + 1
[1,-4,-2,3] => []
=> []
=> []
=> ? = 0 + 1
[1,4,3,2] => []
=> []
=> []
=> ? = 0 + 1
[1,-4,3,-2] => []
=> []
=> []
=> ? = 0 + 1
[2,1,3,4] => []
=> []
=> []
=> ? = 0 + 1
[-2,-1,3,4] => []
=> []
=> []
=> ? = 0 + 1
[2,1,4,3] => []
=> []
=> []
=> ? = 0 + 1
[2,1,-4,-3] => []
=> []
=> []
=> ? = 0 + 1
[-2,-1,4,3] => []
=> []
=> []
=> ? = 0 + 1
[-2,-1,-4,-3] => []
=> []
=> []
=> ? = 0 + 1
[2,3,1,4] => []
=> []
=> []
=> ? = 0 + 1
[2,-3,-1,4] => []
=> []
=> []
=> ? = 0 + 1
[-2,3,-1,4] => []
=> []
=> []
=> ? = 0 + 1
[-2,-3,1,4] => []
=> []
=> []
=> ? = 0 + 1
[2,3,4,1] => []
=> []
=> []
=> ? = 0 + 1
[2,3,-4,-1] => []
=> []
=> []
=> ? = 0 + 1
[2,-3,4,-1] => []
=> []
=> []
=> ? = 0 + 1
[2,-3,-4,1] => []
=> []
=> []
=> ? = 0 + 1
[-2,3,4,-1] => []
=> []
=> []
=> ? = 0 + 1
[-2,3,-4,1] => []
=> []
=> []
=> ? = 0 + 1
Description
Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra.
Matching statistic: St000744
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[1] => []
=> []
=> ?
=> ? = 0
[-1] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[1,2] => []
=> []
=> ?
=> ? = 0
[1,-2] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[-1,2] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[2,1] => []
=> []
=> ?
=> ? = 0
[2,-1] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[-2,1] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[-2,-1] => []
=> []
=> ?
=> ? = 0
[1,2,3] => []
=> []
=> ?
=> ? = 0
[1,2,-3] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[1,-2,3] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[-1,2,3] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 3
[1,3,2] => []
=> []
=> ?
=> ? = 0
[1,3,-2] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[1,-3,2] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[1,-3,-2] => []
=> []
=> ?
=> ? = 0
[-1,3,2] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[-1,-3,-2] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[2,1,3] => []
=> []
=> ?
=> ? = 0
[2,1,-3] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[-2,1,3] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[-2,-1,3] => []
=> []
=> ?
=> ? = 0
[-2,-1,-3] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[2,3,1] => []
=> []
=> ?
=> ? = 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[2,-3,-1] => []
=> []
=> ?
=> ? = 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[-2,3,-1] => []
=> []
=> ?
=> ? = 0
[-2,-3,1] => []
=> []
=> ?
=> ? = 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[3,1,2] => []
=> []
=> ?
=> ? = 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[3,-1,-2] => []
=> []
=> ?
=> ? = 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[-3,1,-2] => []
=> []
=> ?
=> ? = 0
[-3,-1,2] => []
=> []
=> ?
=> ? = 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[3,2,1] => []
=> []
=> ?
=> ? = 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[3,-2,1] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[-3,2,1] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[-3,2,-1] => []
=> []
=> ?
=> ? = 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[-3,-2,-1] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[1,2,3,4] => []
=> []
=> ?
=> ? = 0
[1,2,3,-4] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[1,2,-3,4] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[1,-2,3,4] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 3
[-1,2,3,4] => [1]
=> [1,0]
=> [[1],[2]]
=> 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 3
[1,2,4,3] => []
=> []
=> ?
=> ? = 0
[1,2,-4,-3] => []
=> []
=> ?
=> ? = 0
[1,3,2,4] => []
=> []
=> ?
=> ? = 0
[1,-3,-2,4] => []
=> []
=> ?
=> ? = 0
[1,3,4,2] => []
=> []
=> ?
=> ? = 0
[1,3,-4,-2] => []
=> []
=> ?
=> ? = 0
[1,-3,4,-2] => []
=> []
=> ?
=> ? = 0
[1,-3,-4,2] => []
=> []
=> ?
=> ? = 0
[1,4,2,3] => []
=> []
=> ?
=> ? = 0
[1,4,-2,-3] => []
=> []
=> ?
=> ? = 0
[1,-4,2,-3] => []
=> []
=> ?
=> ? = 0
[1,-4,-2,3] => []
=> []
=> ?
=> ? = 0
[1,4,3,2] => []
=> []
=> ?
=> ? = 0
[1,-4,3,-2] => []
=> []
=> ?
=> ? = 0
[2,1,3,4] => []
=> []
=> ?
=> ? = 0
[-2,-1,3,4] => []
=> []
=> ?
=> ? = 0
[2,1,4,3] => []
=> []
=> ?
=> ? = 0
[2,1,-4,-3] => []
=> []
=> ?
=> ? = 0
[-2,-1,4,3] => []
=> []
=> ?
=> ? = 0
[-2,-1,-4,-3] => []
=> []
=> ?
=> ? = 0
[2,3,1,4] => []
=> []
=> ?
=> ? = 0
[2,-3,-1,4] => []
=> []
=> ?
=> ? = 0
[-2,3,-1,4] => []
=> []
=> ?
=> ? = 0
[-2,-3,1,4] => []
=> []
=> ?
=> ? = 0
[2,3,4,1] => []
=> []
=> ?
=> ? = 0
[2,3,-4,-1] => []
=> []
=> ?
=> ? = 0
[2,-3,4,-1] => []
=> []
=> ?
=> ? = 0
[2,-3,-4,1] => []
=> []
=> ?
=> ? = 0
[-2,3,4,-1] => []
=> []
=> ?
=> ? = 0
[-2,3,-4,1] => []
=> []
=> ?
=> ? = 0
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St000044
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000044: Perfect matchings ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000044: Perfect matchings ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[1] => []
=> []
=> []
=> ? = 0 + 1
[-1] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[1,2] => []
=> []
=> []
=> ? = 0 + 1
[1,-2] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[-1,2] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[2,1] => []
=> []
=> []
=> ? = 0 + 1
[2,-1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 3 = 2 + 1
[-2,1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 3 = 2 + 1
[-2,-1] => []
=> []
=> []
=> ? = 0 + 1
[1,2,3] => []
=> []
=> []
=> ? = 0 + 1
[1,2,-3] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[1,-2,3] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[-1,2,3] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 4 = 3 + 1
[1,3,2] => []
=> []
=> []
=> ? = 0 + 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 3 = 2 + 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 3 = 2 + 1
[1,-3,-2] => []
=> []
=> []
=> ? = 0 + 1
[-1,3,2] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 3 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 3 + 1
[-1,-3,-2] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[2,1,3] => []
=> []
=> []
=> ? = 0 + 1
[2,1,-3] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 3 = 2 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 3 + 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 3 = 2 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 3 + 1
[-2,-1,3] => []
=> []
=> []
=> ? = 0 + 1
[-2,-1,-3] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[2,3,1] => []
=> []
=> []
=> ? = 0 + 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 3 + 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 3 + 1
[2,-3,-1] => []
=> []
=> []
=> ? = 0 + 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 3 + 1
[-2,3,-1] => []
=> []
=> []
=> ? = 0 + 1
[-2,-3,1] => []
=> []
=> []
=> ? = 0 + 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 3 + 1
[3,1,2] => []
=> []
=> []
=> ? = 0 + 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 3 + 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 3 + 1
[3,-1,-2] => []
=> []
=> []
=> ? = 0 + 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 3 + 1
[-3,1,-2] => []
=> []
=> []
=> ? = 0 + 1
[-3,-1,2] => []
=> []
=> []
=> ? = 0 + 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 3 + 1
[3,2,1] => []
=> []
=> []
=> ? = 0 + 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 3 = 2 + 1
[3,-2,1] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 3 + 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 3 = 2 + 1
[-3,2,-1] => []
=> []
=> []
=> ? = 0 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 3 + 1
[-3,-2,-1] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[1,2,3,4] => []
=> []
=> []
=> ? = 0 + 1
[1,2,3,-4] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[1,2,-3,4] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[1,-2,3,4] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 4 = 3 + 1
[-1,2,3,4] => [1]
=> [1,0]
=> [(1,2)]
=> 2 = 1 + 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 3 = 2 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 4 = 3 + 1
[1,2,4,3] => []
=> []
=> []
=> ? = 0 + 1
[1,2,-4,-3] => []
=> []
=> []
=> ? = 0 + 1
[1,3,2,4] => []
=> []
=> []
=> ? = 0 + 1
[1,-3,-2,4] => []
=> []
=> []
=> ? = 0 + 1
[1,3,4,2] => []
=> []
=> []
=> ? = 0 + 1
[1,3,-4,-2] => []
=> []
=> []
=> ? = 0 + 1
[1,-3,4,-2] => []
=> []
=> []
=> ? = 0 + 1
[1,-3,-4,2] => []
=> []
=> []
=> ? = 0 + 1
[1,4,2,3] => []
=> []
=> []
=> ? = 0 + 1
[1,4,-2,-3] => []
=> []
=> []
=> ? = 0 + 1
[1,-4,2,-3] => []
=> []
=> []
=> ? = 0 + 1
[1,-4,-2,3] => []
=> []
=> []
=> ? = 0 + 1
[1,4,3,2] => []
=> []
=> []
=> ? = 0 + 1
[1,-4,3,-2] => []
=> []
=> []
=> ? = 0 + 1
[2,1,3,4] => []
=> []
=> []
=> ? = 0 + 1
[-2,-1,3,4] => []
=> []
=> []
=> ? = 0 + 1
[2,1,4,3] => []
=> []
=> []
=> ? = 0 + 1
[2,1,-4,-3] => []
=> []
=> []
=> ? = 0 + 1
[-2,-1,4,3] => []
=> []
=> []
=> ? = 0 + 1
[-2,-1,-4,-3] => []
=> []
=> []
=> ? = 0 + 1
[2,3,1,4] => []
=> []
=> []
=> ? = 0 + 1
[2,-3,-1,4] => []
=> []
=> []
=> ? = 0 + 1
[-2,3,-1,4] => []
=> []
=> []
=> ? = 0 + 1
[-2,-3,1,4] => []
=> []
=> []
=> ? = 0 + 1
[2,3,4,1] => []
=> []
=> []
=> ? = 0 + 1
[2,3,-4,-1] => []
=> []
=> []
=> ? = 0 + 1
[2,-3,4,-1] => []
=> []
=> []
=> ? = 0 + 1
[2,-3,-4,1] => []
=> []
=> []
=> ? = 0 + 1
[-2,3,4,-1] => []
=> []
=> []
=> ? = 0 + 1
[-2,3,-4,1] => []
=> []
=> []
=> ? = 0 + 1
Description
The number of vertices of the unicellular map given by a perfect matching.
If the perfect matching of $2n$ elements is viewed as a fixed point-free involution $\epsilon$ This statistic is counting the number of cycles of the permutation $\gamma \circ \epsilon$ where $\gamma$ is the long cycle $(1,2,3,\ldots,2n)$.
'''Example'''
The perfect matching $[(1,3),(2,4)]$ corresponds to the permutation in $S_4$ with disjoint cycle decomposition $(1,3)(2,4)$. Then the permutation $(1,2,3,4)\circ (1,3)(2,4) = (1,4,3,2)$ has only one cycle.
Let $\epsilon_v(n)$ is the number of matchings of $2n$ such that yield $v$ cycles in the process described above. Harer and Zagier [1] gave the following expression for the generating series of the numbers $\epsilon_v(n)$.
$$
\sum_{v=1}^{n+1} \epsilon_{v}(n) N^v = (2n-1)!! \sum_{k\geq 0}^n \binom{N}{k+1}\binom{n}{k}2^k.
$$
Matching statistic: St001582
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 50%
Values
[1] => []
=> []
=> [1] => ? = 0
[-1] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,2] => []
=> []
=> [1] => ? = 0
[1,-2] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,2] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[2,1] => []
=> []
=> [1] => ? = 0
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[-2,-1] => []
=> []
=> [1] => ? = 0
[1,2,3] => []
=> []
=> [1] => ? = 0
[1,2,-3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,-2,3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,2,3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3
[1,3,2] => []
=> []
=> [1] => ? = 0
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,-3,-2] => []
=> []
=> [1] => ? = 0
[-1,3,2] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-1,-3,-2] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,1,3] => []
=> []
=> [1] => ? = 0
[2,1,-3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-2,-1,3] => []
=> []
=> [1] => ? = 0
[-2,-1,-3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,3,1] => []
=> []
=> [1] => ? = 0
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[2,-3,-1] => []
=> []
=> [1] => ? = 0
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[-2,3,-1] => []
=> []
=> [1] => ? = 0
[-2,-3,1] => []
=> []
=> [1] => ? = 0
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[3,1,2] => []
=> []
=> [1] => ? = 0
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[3,-1,-2] => []
=> []
=> [1] => ? = 0
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[-3,1,-2] => []
=> []
=> [1] => ? = 0
[-3,-1,2] => []
=> []
=> [1] => ? = 0
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[3,2,1] => []
=> []
=> [1] => ? = 0
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[3,-2,1] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[-3,2,-1] => []
=> []
=> [1] => ? = 0
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-3,-2,-1] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,2,3,4] => []
=> []
=> [1] => ? = 0
[1,2,3,-4] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,2,-3,4] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,-2,3,4] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3
[-1,2,3,4] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 4
[1,2,4,3] => []
=> []
=> [1] => ? = 0
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,2,-4,-3] => []
=> []
=> [1] => ? = 0
[1,-2,4,3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,-2,-4,-3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,2,4,3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[-1,2,-4,-3] => [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 4
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 4
[1,3,2,4] => []
=> []
=> [1] => ? = 0
[1,-3,-2,4] => []
=> []
=> [1] => ? = 0
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 4
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 4
[1,3,4,2] => []
=> []
=> [1] => ? = 0
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[1,3,-4,-2] => []
=> []
=> [1] => ? = 0
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[1,-3,4,-2] => []
=> []
=> [1] => ? = 0
[1,-3,-4,2] => []
=> []
=> [1] => ? = 0
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St000782
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%
Values
[1] => []
=> []
=> []
=> ? = 0 - 2
[-1] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[1,2] => []
=> []
=> []
=> ? = 0 - 2
[1,-2] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[-1,2] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 2
[2,1] => []
=> []
=> []
=> ? = 0 - 2
[2,-1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 2 - 2
[-2,1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 2 - 2
[-2,-1] => []
=> []
=> []
=> ? = 0 - 2
[1,2,3] => []
=> []
=> []
=> ? = 0 - 2
[1,2,-3] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[1,-2,3] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 2
[-1,2,3] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 2
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 3 - 2
[1,3,2] => []
=> []
=> []
=> ? = 0 - 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 2 - 2
[1,-3,2] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 2 - 2
[1,-3,-2] => []
=> []
=> []
=> ? = 0 - 2
[-1,3,2] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-1,-3,-2] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[2,1,3] => []
=> []
=> []
=> ? = 0 - 2
[2,1,-3] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[2,-1,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 2 - 2
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 2 - 2
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-2,-1,3] => []
=> []
=> []
=> ? = 0 - 2
[-2,-1,-3] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[2,3,1] => []
=> []
=> []
=> ? = 0 - 2
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[2,-3,-1] => []
=> []
=> []
=> ? = 0 - 2
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[-2,3,-1] => []
=> []
=> []
=> ? = 0 - 2
[-2,-3,1] => []
=> []
=> []
=> ? = 0 - 2
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[3,1,2] => []
=> []
=> []
=> ? = 0 - 2
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[3,-1,-2] => []
=> []
=> []
=> ? = 0 - 2
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[-3,1,-2] => []
=> []
=> []
=> ? = 0 - 2
[-3,-1,2] => []
=> []
=> []
=> ? = 0 - 2
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[3,2,1] => []
=> []
=> []
=> ? = 0 - 2
[3,2,-1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 2 - 2
[3,-2,1] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 2 - 2
[-3,2,-1] => []
=> []
=> []
=> ? = 0 - 2
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-3,-2,-1] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[1,2,3,4] => []
=> []
=> []
=> ? = 0 - 2
[1,2,3,-4] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[1,2,-3,4] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 2
[1,-2,3,4] => [1]
=> [1,0]
=> [(1,2)]
=> ? = 1 - 2
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 2
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 2 - 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 3 - 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 3 - 2
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 3 - 2
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1 = 3 - 2
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-1,-3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[1,4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[1,4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[1,-4,2,3] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[1,-4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[1,4,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[1,-4,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-1,4,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-1,-4,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[2,-1,3,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[2,-1,-3,4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-2,1,3,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[-2,1,-3,4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 3 - 2
[2,-1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 3 - 2
[2,-1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 3 - 2
[-2,1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 3 - 2
[-2,1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 3 - 2
[2,3,-1,4] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[2,-3,1,4] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
[-2,3,1,4] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 3 - 2
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001491
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 17%
Values
[1] => []
=> []
=> => ? = 0 - 1
[-1] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,2] => []
=> []
=> => ? = 0 - 1
[1,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[2,1] => []
=> []
=> => ? = 0 - 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[-2,-1] => []
=> []
=> => ? = 0 - 1
[1,2,3] => []
=> []
=> => ? = 0 - 1
[1,2,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-2,3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,2,3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[1,3,2] => []
=> []
=> => ? = 0 - 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[1,-3,-2] => []
=> []
=> => ? = 0 - 1
[-1,3,2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[-1,-3,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,1,3] => []
=> []
=> => ? = 0 - 1
[2,1,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[-2,-1,3] => []
=> []
=> => ? = 0 - 1
[-2,-1,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,3,1] => []
=> []
=> => ? = 0 - 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,-3,-1] => []
=> []
=> => ? = 0 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,3,-1] => []
=> []
=> => ? = 0 - 1
[-2,-3,1] => []
=> []
=> => ? = 0 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,1,2] => []
=> []
=> => ? = 0 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,-1,-2] => []
=> []
=> => ? = 0 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-3,1,-2] => []
=> []
=> => ? = 0 - 1
[-3,-1,2] => []
=> []
=> => ? = 0 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,2,1] => []
=> []
=> => ? = 0 - 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[3,-2,1] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[-3,2,-1] => []
=> []
=> => ? = 0 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[-3,-2,-1] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,2,3,4] => []
=> []
=> => ? = 0 - 1
[1,2,3,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,2,-3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,-2,3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,2,3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-2,4,3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-2,-4,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,2,4,3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,2,-4,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,3,2,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-3,-2,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,3,2,4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,-3,-2,4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,3,4,2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,3,-4,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,-3,4,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,-3,-4,2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,4,2,3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,4,-2,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,-4,2,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,-4,-2,3] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,4,-3,2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[1,-4,-3,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,4,3,2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-1,-4,3,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,1,3,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,1,-3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,-1,3,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,-1,-3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,3,1,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,-3,-1,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,3,-1,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,-3,1,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,4,-3,1] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[2,-4,-3,-1] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,4,-3,-1] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[-2,-4,-3,1] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,1,2,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[3,-1,-2,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
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