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Your data matches 141 different statistics following compositions of up to 3 maps.
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Matching statistic: St001629
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Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => 0
[[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => 0
[[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0
[[1,3,5,6],[2,4]]
=> [2,2,2] => [3] => 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [3] => 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [3] => 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [3] => 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [3] => 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [3] => 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [3] => 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [3] => 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [3] => 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [2,1] => 0
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => 1
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [2,1] => 0
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => 1
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [2,1] => 0
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => 1
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [2,1] => 0
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => 1
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [2,1] => 0
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000642
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Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000642: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000642: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[1,3,5],[2],[4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[1,3],[2,5],[4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[1,3,5,6],[2,4]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,3,4,6],[2,5]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[[1,2,4,6],[3,5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,3,5],[2,4,6]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,3,4],[2,5,6]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[[1,2,4],[3,5,6]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 0 + 2
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 0 + 2
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 0 + 2
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,4,6,7],[2,5],[3]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 0 + 2
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,2,6,7],[3,4],[5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,7],[3,6],[5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 0 + 2
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,6],[3,7],[5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,3,5,6],[2,4],[7]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 0 + 2
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,5,7],[2],[4],[6]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 2
[[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 0 + 2
[[1,3,5,6],[2],[4],[7]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 0 + 2
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 1 + 2
[[1,3,5],[2,6,7],[4]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,3,7],[2,5],[4,6]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,3,5],[2,4],[6,7]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,3,7],[2,5],[4],[6]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,3,5],[2,7],[4],[6]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
[[1,3],[2,5],[4,7],[6]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> 2 = 0 + 2
Description
The size of the smallest orbit of antichains under Panyushev complementation.
Matching statistic: St001667
(load all 108 compositions to match this statistic)
(load all 108 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001667: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001667: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 0 + 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => 2 = 0 + 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [4,2,1,5,3] => 2 = 0 + 2
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 3 = 1 + 2
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,1,3,5,4,6] => [2,1,5,3,4,6] => 3 = 1 + 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,3,2,5,4,6] => [3,1,2,5,4,6] => 3 = 1 + 2
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => [4,2,1,3,5,6] => 3 = 1 + 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,1,5,3,4,6] => [2,1,3,5,4,6] => 3 = 1 + 2
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,3,5,2,4,6] => [3,1,5,2,4,6] => 3 = 1 + 2
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3 = 1 + 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [2,1,3,5,6,4] => [2,1,5,3,6,4] => 3 = 1 + 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,3,2,5,6,4] => [3,1,2,5,6,4] => 3 = 1 + 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [3,2,1,5,4,6] => [2,3,1,5,4,6] => 3 = 1 + 2
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => [4,2,1,5,3,6] => 3 = 1 + 2
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [2,1,5,4,3,6] => [2,1,4,5,3,6] => 3 = 1 + 2
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [1,3,5,4,2,6] => [3,1,5,4,2,6] => 3 = 1 + 2
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,3,6,5] => [4,2,1,3,6,5] => 3 = 1 + 2
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [2,1,5,3,6,4] => [2,1,3,5,6,4] => 3 = 1 + 2
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [1,3,5,2,6,4] => [3,1,5,2,6,4] => 3 = 1 + 2
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,1,4,6,3,5] => [2,1,6,4,3,5] => 3 = 1 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [3,5,2,1,4,6] => [5,3,2,1,4,6] => 3 = 1 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [2,5,4,1,3,6] => [5,2,4,1,3,6] => 3 = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => [4,2,6,1,3,5] => 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [2,4,1,6,5,3] => [4,2,1,6,5,3] => 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,3,5,6,4,2] => [3,1,5,6,4,2] => 3 = 1 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,5,2,1,6,4] => [5,3,2,1,6,4] => 3 = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => [5,2,4,1,6,3] => 3 = 1 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,6,1,5,3] => [4,2,6,1,5,3] => 3 = 1 + 2
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 0 + 2
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => [2,1,5,3,4,6,7] => ? = 1 + 2
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => [3,1,2,5,4,6,7] => ? = 0 + 2
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 1 + 2
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [1,3,2,4,6,5,7] => [3,1,2,4,6,5,7] => ? = 0 + 2
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => 3 = 1 + 2
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => ? = 0 + 2
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => [2,1,3,5,4,6,7] => ? = 1 + 2
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,3,5,2,4,6,7] => [3,1,5,2,4,6,7] => ? = 0 + 2
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [2,1,3,6,4,5,7] => [2,1,6,3,4,5,7] => ? = 1 + 2
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [1,3,2,6,4,5,7] => [3,1,2,6,4,5,7] => ? = 0 + 2
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => 3 = 1 + 2
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 0 + 2
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [1,3,4,2,6,5,7] => [3,1,4,2,6,5,7] => ? = 0 + 2
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [2,1,3,5,6,4,7] => [2,1,5,3,6,4,7] => ? = 1 + 2
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [1,3,2,5,6,4,7] => [3,1,2,5,6,4,7] => ? = 0 + 2
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [2,1,4,3,5,7,6] => [2,1,4,3,7,5,6] => ? = 0 + 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => [2,1,5,3,4,7,6] => ? = 1 + 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,3,2,5,4,7,6] => [3,1,2,5,4,7,6] => ? = 0 + 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => ? = 1 + 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [1,3,2,4,6,7,5] => [3,1,2,4,6,7,5] => ? = 0 + 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => 3 = 1 + 2
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,1,5,4,6,7] => [2,3,1,5,4,6,7] => ? = 0 + 2
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => [4,2,1,5,3,6,7] => ? = 0 + 2
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [2,1,5,4,3,6,7] => [2,1,4,5,3,6,7] => ? = 1 + 2
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [1,3,5,4,2,6,7] => [3,1,5,4,2,6,7] => ? = 0 + 2
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [3,2,1,4,6,5,7] => [2,3,1,4,6,5,7] => ? = 0 + 2
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => ? = 0 + 2
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [1,4,3,2,6,5,7] => [4,1,3,2,6,5,7] => ? = 1 + 2
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [2,1,5,3,6,4,7] => [2,1,3,5,6,4,7] => ? = 1 + 2
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [1,3,5,2,6,4,7] => [3,1,5,2,6,4,7] => ? = 0 + 2
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => ? = 0 + 2
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [1,3,4,6,2,5,7] => [3,1,4,6,2,5,7] => ? = 0 + 2
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [2,1,3,6,5,4,7] => [2,1,6,3,5,4,7] => ? = 1 + 2
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [1,3,2,6,5,4,7] => [3,1,2,6,5,4,7] => ? = 0 + 2
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [1,2,4,6,5,3,7] => [1,2,6,4,5,3,7] => 3 = 1 + 2
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [2,4,1,3,5,7,6] => [4,2,1,3,5,7,6] => ? = 0 + 2
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [2,1,5,3,4,7,6] => [2,1,3,5,4,7,6] => ? = 1 + 2
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [1,3,5,2,4,7,6] => [3,1,5,2,4,7,6] => ? = 0 + 2
[[1,3,4,5],[2,7],[6]]
=> [6,2,7,1,3,4,5] => [2,1,3,6,4,7,5] => [2,1,6,3,4,7,5] => ? = 1 + 2
[[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => [1,3,2,6,4,7,5] => [3,1,2,6,4,7,5] => ? = 0 + 2
[[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [1,2,4,6,3,7,5] => [1,2,6,4,3,7,5] => 3 = 1 + 2
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,5,7,6] => ? = 0 + 2
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => [2,1,5,3,7,4,6] => ? = 1 + 2
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [1,3,2,5,7,4,6] => [3,1,2,5,7,4,6] => ? = 0 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,2,1,4,6,7] => [5,3,2,1,4,6,7] => ? = 0 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,4,1,3,6,7] => [5,2,4,1,3,6,7] => ? = 1 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [3,2,6,1,4,5,7] => [2,3,6,1,4,5,7] => ? = 0 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,1,3,5,7] => [4,2,6,1,3,5,7] => ? = 0 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [1,4,6,3,2,5,7] => [4,1,6,3,2,5,7] => ? = 1 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [2,1,6,5,3,4,7] => [2,1,5,6,3,4,7] => ? = 1 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [1,3,6,5,2,4,7] => [3,1,6,5,2,4,7] => ? = 0 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => [4,2,1,7,3,5,6] => ? = 0 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,1,5,7,3,4,6] => [2,1,7,5,3,4,6] => ? = 1 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => [3,1,5,7,2,4,6] => ? = 0 + 2
[[1,2,3],[4,5,7],[6]]
=> [6,4,5,7,1,2,3] => [1,2,4,6,5,7,3] => [1,2,6,4,5,7,3] => 3 = 1 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,2,4,6,7,5,3] => [1,2,6,4,7,5,3] => 3 = 1 + 2
Description
The maximal size of a pair of weak twins for a permutation.
A pair of weak twins in a permutation is a pair of two disjoint subsequences of the same length with the same descent pattern. More formally, a pair of weak twins of size $k$ for a permutation $\pi$ of length $n$ are two disjoint lists $1 \leq i_1 < \dots < i_k \leq n$ and $1 \leq j_1 < \dots < j_k \leq n$ such that $\pi(i_a) < \pi(i_{a+1})$ if and only if $\pi(j_a) < \pi(j_{a+1})$ for all $1 \leq a < k$.
Matching statistic: St001958
(load all 293 compositions to match this statistic)
(load all 293 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => 4 = 0 + 4
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,5,3,2,4] => 4 = 0 + 4
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,5,4,2,1] => [2,4,3,5,1] => 4 = 0 + 4
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,3,2,4,5,1] => [5,3,6,2,4,1] => 5 = 1 + 4
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [6,4,2,3,5,1] => [5,4,2,6,3,1] => 5 = 1 + 4
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [2,4,6,3,5,1] => [5,2,6,4,3,1] => 5 = 1 + 4
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => 5 = 1 + 4
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => 5 = 1 + 4
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [4,2,6,3,5,1] => [5,6,4,2,3,1] => 5 = 1 + 4
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,3,5,2,4,1] => [4,5,6,3,2,1] => 5 = 1 + 4
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,4,5,2,3,1] => [3,5,2,6,4,1] => 5 = 1 + 4
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [2,4,5,6,3,1] => [3,2,6,4,5,1] => 5 = 1 + 4
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,6,4,3,5,1] => [5,2,4,6,3,1] => 5 = 1 + 4
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [3,6,4,2,5,1] => [5,4,3,6,2,1] => 5 = 1 + 4
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [4,6,3,2,5,1] => [5,3,6,4,2,1] => 5 = 1 + 4
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [4,2,3,6,5,1] => [5,4,2,3,6,1] => 5 = 1 + 4
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [3,6,5,2,4,1] => [4,5,3,2,6,1] => 5 = 1 + 4
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [4,6,5,2,3,1] => [3,5,2,4,6,1] => 5 = 1 + 4
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [4,2,5,6,3,1] => [3,6,4,2,5,1] => 5 = 1 + 4
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [5,6,3,2,4,1] => [4,3,6,2,5,1] => 5 = 1 + 4
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [1,4,2,6,3,5] => [1,6,4,2,3,5] => 5 = 1 + 4
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [1,4,3,6,2,5] => [1,6,4,3,2,5] => 5 = 1 + 4
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [1,5,3,6,2,4] => [1,6,5,3,2,4] => 5 = 1 + 4
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [3,5,6,4,2,1] => [2,4,3,6,5,1] => 5 = 1 + 4
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [4,5,6,3,2,1] => [2,3,6,4,5,1] => 5 = 1 + 4
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,5,2,3,1,6] => [3,5,2,4,1,6] => 5 = 1 + 4
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [4,2,6,5,3,1] => [3,5,4,2,6,1] => 5 = 1 + 4
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [4,3,6,5,2,1] => [2,5,4,3,6,1] => 5 = 1 + 4
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [5,3,6,4,2,1] => [2,4,6,5,3,1] => 5 = 1 + 4
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [7,3,2,4,5,6,1] => [6,3,7,2,4,5,1] => ? = 0 + 4
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [7,4,2,3,5,6,1] => [6,4,2,7,3,5,1] => ? = 1 + 4
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [2,4,7,3,5,6,1] => [6,2,7,4,3,5,1] => ? = 0 + 4
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [7,5,2,3,4,6,1] => [6,5,2,3,7,4,1] => ? = 1 + 4
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [2,5,7,3,4,6,1] => [6,2,7,3,5,4,1] => ? = 0 + 4
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [3,5,7,2,4,6,1] => [6,7,3,2,5,4,1] => ? = 1 + 4
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 0 + 4
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1 + 4
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [4,2,7,3,5,6,1] => [6,7,4,2,3,5,1] => ? = 0 + 4
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1 + 4
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [5,2,7,3,4,6,1] => [6,7,5,2,3,4,1] => ? = 0 + 4
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [5,3,7,2,4,6,1] => [6,7,5,3,2,4,1] => ? = 1 + 4
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [7,3,5,2,4,6,1] => [6,5,7,3,2,4,1] => ? = 0 + 4
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [2,3,5,7,4,6,1] => [6,2,3,7,5,4,1] => ? = 0 + 4
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [7,4,5,2,3,6,1] => [6,5,2,7,4,3,1] => ? = 1 + 4
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [2,4,5,7,3,6,1] => [6,2,7,4,5,3,1] => ? = 0 + 4
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [7,3,6,2,4,5,1] => [5,6,7,3,2,4,1] => ? = 0 + 4
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [7,4,6,2,3,5,1] => [5,6,2,7,4,3,1] => ? = 1 + 4
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [2,4,6,7,3,5,1] => [5,2,7,4,3,6,1] => ? = 0 + 4
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [7,5,6,2,3,4,1] => [4,6,2,3,7,5,1] => ? = 1 + 4
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [2,5,6,7,3,4,1] => [4,2,7,3,5,6,1] => ? = 0 + 4
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [3,5,6,7,2,4,1] => [4,7,3,2,5,6,1] => ? = 1 + 4
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [2,7,4,3,5,6,1] => [6,2,4,7,3,5,1] => ? = 0 + 4
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [3,7,4,2,5,6,1] => [6,4,3,7,2,5,1] => ? = 0 + 4
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [4,7,3,2,5,6,1] => [6,3,7,4,2,5,1] => ? = 1 + 4
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [4,2,3,7,5,6,1] => [6,4,2,3,7,5,1] => ? = 0 + 4
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [2,7,5,3,4,6,1] => [6,2,5,3,7,4,1] => ? = 0 + 4
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [3,7,5,2,4,6,1] => [6,5,3,2,7,4,1] => ? = 0 + 4
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [3,2,5,7,4,6,1] => [6,3,2,7,5,4,1] => ? = 1 + 4
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [4,7,5,2,3,6,1] => [6,5,2,4,7,3,1] => ? = 1 + 4
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [4,2,5,7,3,6,1] => [6,7,4,2,5,3,1] => ? = 0 + 4
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [5,7,3,2,4,6,1] => [6,3,7,2,5,4,1] => ? = 0 + 4
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [5,2,3,7,4,6,1] => [6,7,2,5,3,4,1] => ? = 0 + 4
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [5,7,4,2,3,6,1] => [6,4,2,7,5,3,1] => ? = 1 + 4
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [5,2,4,7,3,6,1] => [6,7,5,2,4,3,1] => ? = 0 + 4
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [5,3,4,7,2,6,1] => [6,7,5,3,4,2,1] => ? = 1 + 4
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [3,7,6,2,4,5,1] => [5,6,3,2,4,7,1] => ? = 0 + 4
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [4,7,6,2,3,5,1] => [5,6,2,4,3,7,1] => ? = 1 + 4
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [4,2,6,7,3,5,1] => [5,7,4,2,3,6,1] => ? = 0 + 4
[[1,3,4,5],[2,7],[6]]
=> [6,2,7,1,3,4,5] => [5,7,6,2,3,4,1] => [4,6,2,3,5,7,1] => ? = 1 + 4
[[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => [5,2,6,7,3,4,1] => [4,7,5,2,3,6,1] => ? = 0 + 4
[[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [5,3,6,7,2,4,1] => [4,7,5,3,2,6,1] => ? = 1 + 4
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [6,7,3,2,4,5,1] => [5,3,7,2,4,6,1] => ? = 0 + 4
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [6,7,4,2,3,5,1] => [5,4,2,7,3,6,1] => ? = 1 + 4
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [6,2,4,7,3,5,1] => [5,7,6,2,4,3,1] => ? = 0 + 4
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [1,4,2,7,3,5,6] => [1,7,4,2,3,5,6] => ? = 0 + 4
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [1,4,3,7,2,5,6] => [1,7,4,3,2,5,6] => ? = 1 + 4
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [1,5,2,7,3,4,6] => [1,7,5,2,3,4,6] => ? = 0 + 4
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [1,5,3,7,2,4,6] => [1,7,5,3,2,4,6] => ? = 0 + 4
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [5,3,2,7,4,6,1] => [6,3,7,5,2,4,1] => ? = 1 + 4
[[1,5,7],[2],[3],[4],[6]]
=> [6,4,3,2,1,5,7] => [1,5,3,2,7,4,6] => [1,3,7,5,2,4,6] => 5 = 1 + 4
Description
The degree of the polynomial interpolating the values of a permutation.
Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$.
This statistic records the degree of $p$.
Matching statistic: St001008
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,6,7],[3,4],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,7],[3,6],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,7],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6],[2,4],[7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,5,7],[2],[4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
Description
Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001010: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001010: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,6,7],[3,4],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,7],[3,6],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,7],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6],[2,4],[7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,5,7],[2],[4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
Description
Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001017
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001017: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001017: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,6,7],[3,4],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,7],[3,6],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,7],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6],[2,4],[7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,5,7],[2],[4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
Description
Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001164
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001164: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001164: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,6,7],[3,4],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,7],[3,6],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,7],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
[[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[[1,3,5,6],[2,4],[7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
[[1,3,5,7],[2],[4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
Description
Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules.
Matching statistic: St001570
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,6,7],[3,4],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,7],[3,6],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,6],[3,7],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3,5,6],[2,4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2,4,6],[3,5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,3,5,7],[2],[4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001480
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 2
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 2
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 2
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 2
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[[1,4,6,7],[2,5],[3]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,2,6,7],[3,4],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 2
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,2,4,7],[3,6],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 2
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 2
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,2,4,6],[3,7],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 2
[[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[[1,3,5,6],[2,4],[7]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 2
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 2
[[1,3,5,7],[2],[4],[6]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 2
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001517The length of a longest pair of twins in a permutation. St000443The number of long tunnels of a Dyck path. St000829The Ulam distance of a permutation to the identity permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001742The difference of the maximal and the minimal degree in a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000501The size of the first part in the decomposition of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001871The number of triconnected components of a graph. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St000741The Colin de Verdière graph invariant. St000864The number of circled entries of the shifted recording tableau of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001812The biclique partition number of a graph. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000309The number of vertices with even degree. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000822The Hadwiger number of the graph. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000956The maximal displacement of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001391The disjunction number of a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000019The cardinality of the support of a permutation. St000030The sum of the descent differences of a permutations. St000060The greater neighbor of the maximum. St000075The orbit size of a standard tableau under promotion. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000325The width of the tree associated to a permutation. St000384The maximal part of the shifted composition of an integer partition. St000470The number of runs in a permutation. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000795The mad of a permutation. St000957The number of Bruhat lower covers of a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000171The degree of the graph. St000226The convexity of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000471The sum of the ascent tops of a permutation. St000625The sum of the minimal distances to a greater element. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001074The number of inversions of the cyclic embedding of a permutation. St001118The acyclic chromatic index of a graph. St001497The position of the largest weak excedence of a permutation. St001519The pinnacle sum of a permutation. St001725The harmonious chromatic number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001883The mutual visibility number of a graph. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000167The number of leaves of an ordered tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001746The coalition number of a graph. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000301The number of facets of the stable set polytope of a graph. St001430The number of positive entries in a signed permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001638The book thickness of a graph.
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