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Your data matches 304 different statistics following compositions of up to 3 maps.
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Matching statistic: St000408
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(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,4,3] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,3,2,4] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,3,4,2] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,4,1,5,2] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,4,2,3] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,4,2,5,3] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,5,2,1,3] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [5,2,1,3,4] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,4,5,2] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [1,5,3,2,4] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [6,2,3,1,4,5] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,1,5,2,3] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,1,5,2,4] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [1,2,3,6,4,5] => 0
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,6,2,1,3,4] => 0
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [6,2,1,3,4,5] => 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,4,5,2,3] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => 0
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [5,1,6,2,3,4] => 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [1,2,5,6,3,4] => 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [1,2,4,6,3,5] => 0
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,2,3] => 0
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [5,2,1,3,4,6] => 0
Description
The number of occurrences of the pattern 4231 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to be smooth [2].
Matching statistic: St001550
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,1,3,4] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,1,3,4,5] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,1,4,3,5] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [3,4,6,1,5,2] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3,5,1,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,2,5,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [4,6,1,2,5,3] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,4,6,5,1,3] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,3,6,5,2] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,1,5,3,4,6] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,6,3,2,5,4] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,1,4,5,3,6] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,2,4,3] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,6,5,3] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,4,1,2,6,5] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,1,3,6,4,2] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,3,5,1,4] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,6,3,5,4] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,3,4,6,5,2] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,6,3,5] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,2,6,4,5,3] => 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,2,3,5,4] => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 0
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [1,3,5,7,2,6,4] => 0
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [1,4,5,2,7,6,3] => 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,3,4,6,1,5] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,4,2,6,5] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,2,3,6,5] => 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,6,4] => 0
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [1,3,5,2,7,6,4] => 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,3,7,4,2,6,5] => 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [1,4,2,7,3,6,5] => 0
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [1,4,5,3,7,6,2] => 0
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [1,2,3,5,7,4,6] => 0
Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St000232
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> 0
[2]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => {{1,2,3,4,5},{6}}
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => {{1,2,3},{4,5,6}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => {{1},{2,3,4,5,6}}
=> 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}}
=> 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 0
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ?
=> ? = 0
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 0
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ?
=> ? = 0
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 0
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ?
=> ? = 0
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ?
=> ? = 0
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ?
=> ? = 0
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ?
=> ? = 0
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ?
=> ? = 0
[]
=> []
=> [] => {}
=> ? = 0
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000491
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => {{1,2,3,4,5},{6}}
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => {{1,2,3},{4,5,6}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => {{1},{2,3,4,5,6}}
=> 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}}
=> 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 0
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ?
=> ? = 0
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 0
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ?
=> ? = 0
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 0
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ?
=> ? = 0
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ?
=> ? = 0
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ?
=> ? = 0
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ?
=> ? = 0
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ?
=> ? = 0
[]
=> []
=> [] => {}
=> ? = 0
Description
The number of inversions of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$.
This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller".
This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000497
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => {{1,2,3,4,5},{6}}
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => {{1,2,3},{4,5,6}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => {{1},{2,3,4,5,6}}
=> 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}}
=> 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 0
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ?
=> ? = 0
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 0
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ?
=> ? = 0
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 0
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ?
=> ? = 0
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ?
=> ? = 0
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ?
=> ? = 0
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ?
=> ? = 0
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ?
=> ? = 0
[]
=> []
=> [] => {}
=> ? = 0
Description
The lcb statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000562
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000562: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000562: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => {{1,2,3,4,5},{6}}
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => {{1,2,3},{4,5,6}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => {{1},{2,3,4,5,6}}
=> 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}}
=> 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 0
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ?
=> ? = 0
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 0
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ?
=> ? = 0
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 0
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ?
=> ? = 0
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ?
=> ? = 0
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ?
=> ? = 0
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ?
=> ? = 0
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ?
=> ? = 0
[]
=> []
=> [] => {}
=> ? = 0
Description
The number of internal points of a set partition.
An element $e$ is internal, if there are $f < e < g$ such that the blocks of $f$ and $g$ have larger minimal element than the block of $e$. See Section 5.5 of [1]
Matching statistic: St000563
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000563: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000563: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => {{1,2,3,4,5},{6}}
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => {{1,2,3},{4,5,6}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => {{1},{2,3,4,5,6}}
=> 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}}
=> 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 0
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ?
=> ? = 0
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 0
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ?
=> ? = 0
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 0
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ?
=> ? = 0
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ?
=> ? = 0
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ?
=> ? = 0
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ?
=> ? = 0
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ?
=> ? = 0
[]
=> []
=> [] => {}
=> ? = 0
Description
The number of overlapping pairs of blocks of a set partition.
This is also the number of occurrences of the pattern {{1,3},{2,4}} such that 1,2 are minimal and 3,4 are maximal.
Matching statistic: St000565
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000565: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000565: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => {{1,2,3,4,5},{6}}
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => {{1,2,3},{4,5,6}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => {{1},{2,3,4,5,6}}
=> 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}}
=> 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 0
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ?
=> ? = 0
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 0
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ?
=> ? = 0
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 0
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ?
=> ? = 0
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ?
=> ? = 0
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ?
=> ? = 0
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ?
=> ? = 0
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ?
=> ? = 0
[]
=> []
=> [] => {}
=> ? = 0
Description
The major index of a set partition.
Let $\pi=B_1/B_2/\dots/B_k$ with $\min B_1<\min B_2<\dots<\min B_k$ a set partition. Let $d_i$ be the number of elements in $B_i$ larger than $\min B_{i+1}$. Then the major index of $\pi$ is $1d_1+2d_2+\dots+(k-1)d_{k-1}$.
Matching statistic: St000581
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => {{1,2,3,4,5},{6}}
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => {{1,2,3},{4,5,6}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => {{1},{2,3,4,5,6}}
=> 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}}
=> 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 0
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ?
=> ? = 0
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 0
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ?
=> ? = 0
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 0
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ?
=> ? = 0
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ?
=> ? = 0
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ?
=> ? = 0
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ?
=> ? = 0
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ?
=> ? = 0
[]
=> []
=> [] => {}
=> ? = 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal.
Matching statistic: St000582
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000582: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000582: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => {{1,2,3,4,5},{6}}
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => {{1,2,3},{4,5,6}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => {{1},{2,3,4,5,6}}
=> 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}}
=> 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 0
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ?
=> ? = 0
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 0
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ?
=> ? = 0
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 0
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,6,7,8,5] => {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ?
=> ? = 0
[5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ?
=> ? = 0
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ?
=> ? = 0
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ?
=> ? = 0
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ?
=> ? = 0
[]
=> []
=> [] => {}
=> ? = 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block.
The following 294 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St001513The number of nested exceedences of a permutation. St001344The neighbouring number of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000667The greatest common divisor of the parts of the partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000878The number of ones minus the number of zeros of a binary word. St000627The exponent of a binary word. St001434The number of negative sum pairs of a signed permutation. St000264The girth of a graph, which is not a tree. St000666The number of right tethers of a permutation. St000664The number of right ropes of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001162The minimum jump of a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000842The breadth of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001722The number of minimal chains with small intervals between a binary word and the top element. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000405The number of occurrences of the pattern 1324 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000516The number of stretching pairs of a permutation. St000629The defect of a binary word. St000787The number of flips required to make a perfect matching noncrossing. St001204Call 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000260The radius of a connected graph. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000570The Edelman-Greene number of a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000297The number of leading ones in a binary word. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000326The position of the first one in a binary word after appending a 1 at the end. St000750The number of occurrences of the pattern 4213 in a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000768The number of peaks in an integer composition. St000764The number of strong records in an integer composition. St000058The order of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000650The number of 3-rises of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001429The number of negative entries in a signed permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001856The number of edges in the reduced word graph of a permutation. St000286The number of connected components of the complement of a graph. St000657The smallest part of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001260The permanent of an alternating sign matrix. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001481The minimal height of a peak of a Dyck path. St001518The number of graphs with the same ordinary spectrum as the given graph. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000807The sum of the heights of the valleys of the associated bargraph. St000894The trace of an alternating sign matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001520The number of strict 3-descents. St001524The degree of symmetry of a binary word. St001556The number of inversions of the third entry of a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001845The number of join irreducibles minus the rank of a lattice. St001867The number of alignments of type EN of a signed permutation. St000295The length of the border of a binary word. St000763The sum of the positions of the strong records of an integer composition. St000805The number of peaks of the associated bargraph. St001267The length of the Lyndon factorization of the binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001884The number of borders of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St000900The minimal number of repetitions of a part in an integer composition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001868The number of alignments of type NE of a signed permutation. St000296The length of the symmetric border of a binary word. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000902 The minimal number of repetitions of an integer composition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001889The size of the connectivity set of a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001621The number of atoms of a lattice. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St000068The number of minimal elements in a poset. St000455The second largest eigenvalue of a graph if it is integral. St001410The minimal entry of a semistandard tableau. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000451The length of the longest pattern of the form k 1 2. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000039The number of crossings of a permutation. St000091The descent variation of a composition. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000462The major index minus the number of excedences of a permutation. St000496The rcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001537The number of cyclic crossings of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001857The number of edges in the reduced word graph of a signed permutation. St000021The number of descents of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of a permutation. St001928The number of non-overlapping descents in a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St000862The number of parts of the shifted shape of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
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