searching the database
Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000492
(load all 503 compositions to match this statistic)
(load all 503 compositions to match this statistic)
St000492: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 1
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 5
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 4
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> 3
{{1},{2,4},{3}}
=> 3
{{1},{2},{3,4}}
=> 3
{{1},{2},{3},{4}}
=> 6
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 3
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 7
{{1,2,4,5},{3}}
=> 2
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 6
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 6
{{1,2,5},{3},{4}}
=> 5
{{1,2},{3,5},{4}}
=> 5
{{1,2},{3},{4,5}}
=> 5
{{1,2},{3},{4},{5}}
=> 9
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 5
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 5
{{1,3,5},{2},{4}}
=> 4
{{1,3},{2,5},{4}}
=> 4
{{1,3},{2},{4,5}}
=> 4
{{1,3},{2},{4},{5}}
=> 8
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 5
Description
The rob 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 '''rob''' (right-opener-bigger) 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 is also the number of occurrences of the pattern {{1}, {2}}, such that 2 is the minimal element of a block.
Matching statistic: St000472
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 95%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 95%
Values
{{1,2}}
=> {{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 1
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [3,2,1] => 0
{{1,2},{3}}
=> {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 2
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 3
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 0
{{1,2,3},{4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => 3
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => 2
{{1,2},{3,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 2
{{1,2},{3},{4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 5
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 4
{{1,4},{2,3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 1
{{1},{2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 4
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 3
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 3
{{1},{2},{3,4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 3
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 6
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,3,2,1] => 4
{{1,2,3,5},{4}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,4,2,1] => 3
{{1,2,3},{4,5}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,2,1] => 3
{{1,2,3},{4},{5}}
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,2,1] => 7
{{1,2,4,5},{3}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,4,3,1] => 2
{{1,2,4},{3,5}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,2,4,3,1] => 2
{{1,2,4},{3},{5}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,3,1] => 6
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,4,1] => 2
{{1,2},{3,4,5}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,3,2,5,1] => 2
{{1,2},{3,4},{5}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => 6
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,4,1] => 5
{{1,2},{3,5},{4}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [3,4,2,5,1] => 5
{{1,2},{3},{4,5}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,5,1] => 5
{{1,2},{3},{4},{5}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,5,1] => 9
{{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,4,3,2] => 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,3,1,4,2] => 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,3,2] => 5
{{1,3,5},{2,4}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,4,1,3,2] => 1
{{1,3},{2,4,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,5,3,2] => 1
{{1,3},{2,4},{5}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,1,3,2] => 5
{{1,3,5},{2},{4}}
=> {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,4,2] => 4
{{1,3},{2,5},{4}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,5,1,4,2] => 4
{{1,3},{2},{4,5}}
=> {{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,5,2] => 4
{{1,3},{2},{4},{5}}
=> {{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,5,2] => 8
{{1,4,5},{2,3}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
{{1,4},{2,3,5}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,4,3] => 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 5
{{1,2,3,4,5,6,7}}
=> {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [7,6,5,4,3,2,1] => ? = 0
{{1,2,3,4,5,6},{7}}
=> {{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [6,7,5,4,3,2,1] => ? = 6
{{1,2,3,4,5,7},{6}}
=> {{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [5,7,6,4,3,2,1] => ? = 5
{{1,2,3,4,5},{6,7}}
=> {{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [6,5,7,4,3,2,1] => ? = 5
{{1,2,3,4,5},{6},{7}}
=> {{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [5,6,7,4,3,2,1] => ? = 11
{{1,2,3,4,6,7},{5}}
=> {{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [4,7,6,5,3,2,1] => ? = 4
{{1,2,3,4,6},{5,7}}
=> {{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [7,4,6,5,3,2,1] => ? = 4
{{1,2,3,4,6},{5},{7}}
=> {{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [4,6,7,5,3,2,1] => ? = 10
{{1,2,3,4,7},{5,6}}
=> {{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [5,4,7,6,3,2,1] => ? = 4
{{1,2,3,4},{5,6,7}}
=> {{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [6,5,4,7,3,2,1] => ? = 4
{{1,2,3,4},{5,6},{7}}
=> {{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [5,4,6,7,3,2,1] => ? = 10
{{1,2,3,4,7},{5},{6}}
=> {{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [4,5,7,6,3,2,1] => ? = 9
{{1,2,3,4},{5,7},{6}}
=> {{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [5,6,4,7,3,2,1] => ? = 9
{{1,2,3,4},{5},{6,7}}
=> {{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [4,6,5,7,3,2,1] => ? = 9
{{1,2,3,4},{5},{6},{7}}
=> {{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [4,5,6,7,3,2,1] => ? = 15
{{1,2,3,5,6,7},{4}}
=> {{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [3,7,6,5,4,2,1] => ? = 3
{{1,2,3,5,6},{4,7}}
=> {{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [7,5,3,6,4,2,1] => ? = 3
{{1,2,3,5,6},{4},{7}}
=> {{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [3,6,7,5,4,2,1] => ? = 9
{{1,2,3,5,7},{4,6}}
=> {{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [7,6,3,5,4,2,1] => ? = 3
{{1,2,3,5},{4,6,7}}
=> {{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [6,3,7,5,4,2,1] => ? = 3
{{1,2,3,5},{4,6},{7}}
=> {{1,2,4,7},{3,5},{6}}
=> [2,4,5,7,3,6,1] => [6,7,3,5,4,2,1] => ? = 9
{{1,2,3,5,7},{4},{6}}
=> {{1,2,4,6,7},{3},{5}}
=> [2,4,3,6,5,7,1] => [3,5,7,6,4,2,1] => ? = 8
{{1,2,3,5},{4,7},{6}}
=> {{1,2,4,7},{3,6},{5}}
=> [2,4,6,7,5,3,1] => [5,7,3,6,4,2,1] => ? = 8
{{1,2,3,5},{4},{6,7}}
=> {{1,2,4,7},{3},{5,6}}
=> [2,4,3,7,6,5,1] => [3,6,5,7,4,2,1] => ? = 8
{{1,2,3,5},{4},{6},{7}}
=> {{1,2,4,7},{3},{5},{6}}
=> [2,4,3,7,5,6,1] => [3,5,6,7,4,2,1] => ? = 14
{{1,2,3,6,7},{4,5}}
=> {{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [4,3,7,6,5,2,1] => ? = 3
{{1,2,3,6},{4,5,7}}
=> {{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [7,4,3,6,5,2,1] => ? = 3
{{1,2,3,6},{4,5},{7}}
=> {{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [4,3,6,7,5,2,1] => ? = 9
{{1,2,3,7},{4,5,6}}
=> {{1,2,6,7},{3,4,5}}
=> [2,6,4,5,3,7,1] => [5,4,3,7,6,2,1] => ? = 3
{{1,2,3},{4,5,6,7}}
=> {{1,2,7},{3,4,5,6}}
=> [2,7,4,5,6,3,1] => [6,5,4,3,7,2,1] => ? = 3
{{1,2,3},{4,5,6},{7}}
=> {{1,2,7},{3,4,5},{6}}
=> [2,7,4,5,3,6,1] => [5,4,3,6,7,2,1] => ? = 9
{{1,2,3,7},{4,5},{6}}
=> {{1,2,6,7},{3,4},{5}}
=> [2,6,4,3,5,7,1] => [4,3,5,7,6,2,1] => ? = 8
{{1,2,3},{4,5,7},{6}}
=> {{1,2,7},{3,4,6},{5}}
=> [2,7,4,6,5,3,1] => [5,6,4,3,7,2,1] => ? = 8
{{1,2,3},{4,5},{6,7}}
=> {{1,2,7},{3,4},{5,6}}
=> [2,7,4,3,6,5,1] => [4,3,6,5,7,2,1] => ? = 8
{{1,2,3},{4,5},{6},{7}}
=> {{1,2,7},{3,4},{5},{6}}
=> [2,7,4,3,5,6,1] => [4,3,5,6,7,2,1] => ? = 14
{{1,2,3,6,7},{4},{5}}
=> {{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [3,4,7,6,5,2,1] => ? = 7
{{1,2,3,6},{4,7},{5}}
=> {{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [4,7,3,6,5,2,1] => ? = 7
{{1,2,3,6},{4},{5,7}}
=> {{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [3,7,4,6,5,2,1] => ? = 7
{{1,2,3,6},{4},{5},{7}}
=> {{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [3,4,6,7,5,2,1] => ? = 13
{{1,2,3,7},{4,6},{5}}
=> {{1,2,6,7},{3,5},{4}}
=> [2,6,5,4,3,7,1] => [4,5,3,7,6,2,1] => ? = 7
{{1,2,3},{4,6,7},{5}}
=> {{1,2,7},{3,5,6},{4}}
=> [2,7,5,4,6,3,1] => [4,6,5,3,7,2,1] => ? = 7
{{1,2,3},{4,6},{5,7}}
=> {{1,2,7},{3,5},{4,6}}
=> [2,7,5,6,3,4,1] => [6,3,5,4,7,2,1] => ? = 7
{{1,2,3},{4,6},{5},{7}}
=> {{1,2,7},{3,5},{4},{6}}
=> [2,7,5,4,3,6,1] => [4,5,3,6,7,2,1] => ? = 13
{{1,2,3,7},{4},{5,6}}
=> {{1,2,6,7},{3},{4,5}}
=> [2,6,3,5,4,7,1] => [3,5,4,7,6,2,1] => ? = 7
{{1,2,3},{4,7},{5,6}}
=> {{1,2,7},{3,6},{4,5}}
=> [2,7,6,5,4,3,1] => [5,4,6,3,7,2,1] => ? = 7
{{1,2,3},{4},{5,6,7}}
=> {{1,2,7},{3},{4,5,6}}
=> [2,7,3,5,6,4,1] => [3,6,5,4,7,2,1] => ? = 7
{{1,2,3},{4},{5,6},{7}}
=> {{1,2,7},{3},{4,5},{6}}
=> [2,7,3,5,4,6,1] => [3,5,4,6,7,2,1] => ? = 13
{{1,2,3,7},{4},{5},{6}}
=> {{1,2,6,7},{3},{4},{5}}
=> [2,6,3,4,5,7,1] => [3,4,5,7,6,2,1] => ? = 12
{{1,2,3},{4,7},{5},{6}}
=> {{1,2,7},{3,6},{4},{5}}
=> [2,7,6,4,5,3,1] => [4,5,6,3,7,2,1] => ? = 12
{{1,2,3},{4},{5,7},{6}}
=> {{1,2,7},{3},{4,6},{5}}
=> [2,7,3,6,5,4,1] => [3,5,6,4,7,2,1] => ? = 12
Description
The sum of the ascent bottoms of a permutation.
Matching statistic: St000154
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 73%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 73%
Values
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [1,3,2] => 2
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [2,3,1] => 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,2,1] => 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 5
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 4
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [2,3,1,4] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [2,3,4,1] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [2,4,3,1] => 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [3,2,1,4] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [3,2,4,1] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [3,4,2,1] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 6
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,4,5,3] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,4,3] => 7
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,3,4,2,5] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,3,5,4,2] => 6
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,4,3,2,5] => 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,4,3,5,2] => 5
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,4,5,3,2] => 5
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,4,3,2] => 9
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [3,5,1,2,4] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => 5
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [4,5,1,2,3] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [5,4,1,2,3] => 5
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => 4
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [4,1,2,5,3] => 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,4,5,3] => 4
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,5,4,3] => 8
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [2,3,1,4,5] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [2,5,1,3,4] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [2,3,1,5,4] => 5
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 6
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 5
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 5
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 11
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 4
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 4
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 10
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 4
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => ? = 4
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,2,3,6,5,7,4] => [1,2,3,5,7,6,4] => ? = 10
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 9
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => ? = 9
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,2,3,5,7,6,4] => [1,2,3,6,7,5,4] => ? = 9
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => ? = 15
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 3
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,2,7,3,5,4,6] => [1,2,5,7,3,4,6] => ? = 3
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 9
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => ? = 3
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,2,7,3,4,6,5] => [1,2,6,7,3,4,5] => ? = 3
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,2,6,3,4,7,5] => [1,2,7,6,3,4,5] => ? = 9
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 8
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,2,7,3,6,4,5] => [1,2,6,3,4,7,5] => ? = 8
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 8
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => ? = 14
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ? = 3
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,2,7,4,3,5,6] => [1,2,4,7,3,5,6] => ? = 3
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => ? = 9
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,2,6,4,5,3,7] => [1,2,4,5,6,3,7] => ? = 3
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,2,7,4,5,6,3] => [1,2,4,5,6,7,3] => ? = 3
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,2,6,4,5,7,3] => [1,2,4,5,7,6,3] => ? = 9
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,2,5,4,6,3,7] => [1,2,4,6,5,3,7] => ? = 8
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,2,7,4,6,5,3] => [1,2,4,6,5,7,3] => ? = 8
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,2,5,4,7,6,3] => [1,2,4,6,7,5,3] => ? = 8
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,2,5,4,6,7,3] => [1,2,4,7,6,5,3] => ? = 14
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ? = 7
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,2,7,5,3,4,6] => [1,2,5,3,7,4,6] => ? = 7
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => ? = 7
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => ? = 13
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,2,6,5,4,3,7] => [1,2,5,4,6,3,7] => ? = 7
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,2,7,5,4,6,3] => [1,2,5,4,6,7,3] => ? = 7
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,2,6,7,4,5,3] => [1,2,7,4,5,6,3] => ? = 7
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,2,6,5,4,7,3] => [1,2,5,4,7,6,3] => ? = 13
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,2,4,6,5,3,7] => [1,2,5,6,4,3,7] => ? = 7
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,2,7,6,5,4,3] => [1,2,5,6,4,7,3] => ? = 7
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,2,4,7,5,6,3] => [1,2,5,6,7,4,3] => ? = 7
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,2,4,6,5,7,3] => [1,2,5,7,6,4,3] => ? = 13
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => ? = 12
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,2,7,5,6,4,3] => [1,2,6,5,4,7,3] => ? = 12
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,2,4,7,6,5,3] => [1,2,6,5,7,4,3] => ? = 12
Description
The sum of the descent bottoms of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see [[St000111]].
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!