searching the database
Your data matches 2 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: St000359
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [3,2,1] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 4
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 0
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000496
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00112: Set partitions —complement⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
St000496: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> {{1}}
=> 0
{{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1},{2,3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2,3}}
=> {{1,2},{3}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 3
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 2
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 0
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 2
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 2
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 4
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 0
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 1
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> 0
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> 0
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 3
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 1
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 0
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
{{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
{{1},{2},{3},{4,8},{5},{6},{7}}
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 3
{{1},{2},{3,4},{5,6},{7},{8}}
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 0
{{1},{2},{3,4,5,8},{6},{7}}
=> {{1,4,5,6},{2},{3},{7},{8}}
=> ? = 6
{{1},{2,3},{4},{5},{6,7},{8}}
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 0
{{1},{2,3},{4,5},{6,7},{8}}
=> {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 0
{{1},{2,4},{3},{5,7},{6},{8}}
=> {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 2
{{1},{2,4,6,8},{3},{5},{7}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 6
{{1},{2,3,4},{5,6,7},{8}}
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 0
{{1},{2,7,8},{3,4},{5,6}}
=> {{1,2,7},{3,4},{5,6},{8}}
=> ? = 2
{{1},{2,3,4,5,8},{6},{7}}
=> {{1,4,5,6,7},{2},{3},{8}}
=> ? = 8
{{1},{2,3,4,5,7,8},{6}}
=> {{1,2,4,5,6,7},{3},{8}}
=> ? = 4
{{1},{2,3,4,5,8},{6,7}}
=> {{1,4,5,6,7},{2,3},{8}}
=> ? = 4
{{1},{2,3,4,5,6,8},{7}}
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 5
{{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0
{{1,2},{3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 0
{{1,2},{3,4},{5,7,8},{6}}
=> {{1,2,4},{3},{5,6},{7,8}}
=> ? = 1
{{1,2},{3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
{{1,2},{3,5,6},{4},{7,8}}
=> {{1,2},{3,4,6},{5},{7,8}}
=> ? = 1
{{1,2},{3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 0
{{1,3},{2},{4},{5},{6,8},{7}}
=> {{1,3},{2},{4},{5},{6,8},{7}}
=> ? = 2
{{1,4},{2},{3},{5,8},{6},{7}}
=> {{1,4},{2},{3},{5,8},{6},{7}}
=> ? = 4
{{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 3
{{1,6},{2},{3},{4},{5},{7},{8}}
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 4
{{1,6},{2},{3},{4},{5},{7,8}}
=> {{1,2},{3,8},{4},{5},{6},{7}}
=> ? = 4
{{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 5
{{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6
{{1,3,4},{2},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6,8},{7}}
=> ? = 1
{{1,3,4},{2},{5,7,8},{6}}
=> {{1,2,4},{3},{5,6,8},{7}}
=> ? = 2
{{1,3,4},{2},{5,8},{6,7}}
=> {{1,4},{2,3},{5,6,8},{7}}
=> ? = 2
{{1,3,4,5,7,8},{2},{6}}
=> {{1,2,4,5,6,8},{3},{7}}
=> ? = 5
{{1,2,3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 0
{{1,2,3},{4},{5},{6,7,8}}
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 0
{{1,2,3},{4,5,6,7,8}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0
{{1,4},{2,3},{5,7,8},{6}}
=> {{1,2,4},{3},{5,8},{6,7}}
=> ? = 2
{{1,8},{2,3},{4},{5},{6},{7}}
=> {{1,8},{2},{3},{4},{5},{6,7}}
=> ? = 5
{{1,6,7},{2,3},{4,5},{8}}
=> {{1},{2,3,8},{4,5},{6,7}}
=> ? = 2
{{1,2,4},{3},{5,6,8},{7}}
=> {{1,3,4},{2},{5,7,8},{6}}
=> ? = 4
{{1,8},{2,4},{3},{5},{6},{7}}
=> {{1,8},{2},{3},{4},{5,7},{6}}
=> ? = 6
{{1,8},{2,5},{3},{4},{6},{7}}
=> {{1,8},{2},{3},{4,7},{5},{6}}
=> ? = 7
{{1,2,4,6,7,8},{3},{5}}
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 5
{{1,2,3,4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 0
{{1,2,3,4},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
{{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
{{1,8},{2,3,4},{5},{6},{7}}
=> {{1,8},{2},{3},{4},{5,6,7}}
=> ? = 4
{{1,2,7,8},{3,4},{5,6}}
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 4
{{1,2,3,4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 0
{{1,2,3,4,5},{6,7,8}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 0
{{1,2,3,4,7},{5},{6},{8}}
=> {{1},{2,5,6,7,8},{3},{4}}
=> ? = 8
Description
The rcs 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 '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
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!