Processing math: 100%

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001050
(load all 87 compositions to match this statistic)
Mapping
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => 1 = 0 + 1
{{1},{2}}
 => 2 = 1 + 1
{{1,2,3}}
 => 1 = 0 + 1
{{1,2},{3}}
 => 2 = 1 + 1
{{1,3},{2}}
 => 2 = 1 + 1
{{1},{2,3}}
 => 1 = 0 + 1
{{1},{2},{3}}
 => 3 = 2 + 1
{{1,2,3,4}}
 => 1 = 0 + 1
{{1,2,3},{4}}
 => 2 = 1 + 1
{{1,2,4},{3}}
 => 2 = 1 + 1
{{1,2},{3,4}}
 => 1 = 0 + 1
{{1,2},{3},{4}}
 => 3 = 2 + 1
{{1,3,4},{2}}
 => 1 = 0 + 1
{{1,3},{2,4}}
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => 3 = 2 + 1
{{1,4},{2,3}}
 => 2 = 1 + 1
{{1},{2,3,4}}
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => 3 = 2 + 1
{{1},{2,4},{3}}
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => 1 = 0 + 1
{{1},{2},{3},{4}}
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => 2 = 1 + 1
{{1,2,3},{4,5}}
 => 1 = 0 + 1
{{1,2,3},{4},{5}}
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => 3 = 2 + 1
{{1,2,5},{3,4}}
 => 2 = 1 + 1
{{1,2},{3,4,5}}
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => 3 = 2 + 1
{{1,2},{3,5},{4}}
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => 1 = 0 + 1
{{1,2},{3},{4},{5}}
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => 2 = 1 + 1
{{1,3},{2,4,5}}
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => 3 = 2 + 1
{{1,3,5},{2},{4}}
 => 2 = 1 + 1
{{1,3},{2,5},{4}}
 => 3 = 2 + 1
{{1,3},{2},{4,5}}
 => 1 = 0 + 1
{{1,3},{2},{4},{5}}
 => 4 = 3 + 1
{{1,4,5},{2,3}}
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => 3 = 2 + 1
Description
The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000745
(load all 23 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => [[1,2]]
 => 1 = 0 + 1
{{1},{2}}
 => {{1},{2}}
 => [[1],[2]]
 => 2 = 1 + 1
{{1,2,3}}
 => {{1,2,3}}
 => [[1,2,3]]
 => 1 = 0 + 1
{{1,2},{3}}
 => {{1},{2,3}}
 => [[1,3],[2]]
 => 2 = 1 + 1
{{1,3},{2}}
 => {{1,3},{2}}
 => [[1,3],[2]]
 => 2 = 1 + 1
{{1},{2,3}}
 => {{1,2},{3}}
 => [[1,2],[3]]
 => 1 = 0 + 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [[1],[2],[3]]
 => 3 = 2 + 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [[1,2,3,4]]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 2 = 1 + 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 1 = 0 + 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [[1,3],[2,4]]
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
{{1,4},{2,3}}
 => {{1,4},{2,3}}
 => [[1,3],[2,4]]
 => 2 = 1 + 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 1 = 0 + 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => {{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => 1 = 0 + 1
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => {{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => {{1},{2,4,5},{3}}
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
{{1,2,5},{3,4}}
 => {{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => 2 = 1 + 1
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => [[1,3],[2,5],[4]]
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => {{1,4,5},{2},{3}}
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
{{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => 1 = 0 + 1
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => {{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => [[1,3,5],[2],[4]]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => {{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => 2 = 1 + 1
{{1,3},{2,4,5}}
 => {{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => [[1,4],[2,5],[3]]
 => 3 = 2 + 1
{{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => 2 = 1 + 1
{{1,3},{2,5},{4}}
 => {{1,4},{2},{3,5}}
 => [[1,4],[2,5],[3]]
 => 3 = 2 + 1
{{1,3},{2},{4,5}}
 => {{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => 1 = 0 + 1
{{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => {{1},{2,5},{3,4}}
 => [[1,4],[2,5],[3]]
 => 3 = 2 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => [[1,2]]
 => 0 => 0
{{1},{2}}
 => {{1},{2}}
 => [[1],[2]]
 => 1 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [[1,2,3]]
 => 00 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => [[1,3],[2]]
 => 10 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => [[1,3],[2]]
 => 10 => 1
{{1},{2,3}}
 => {{1,2},{3}}
 => [[1,2],[3]]
 => 01 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [[1],[2],[3]]
 => 11 => 2
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [[1,2,3,4]]
 => 000 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 100 => 1
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 100 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 010 => 0
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 110 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 010 => 0
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [[1,3],[2,4]]
 => 101 => 1
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 110 => 2
{{1,4},{2,3}}
 => {{1,4},{2,3}}
 => [[1,3],[2,4]]
 => 101 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 001 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 101 => 1
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 110 => 2
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 101 => 1
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 011 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 111 => 3
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => 0000 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => [[1,3,4,5],[2]]
 => 1000 => 1
{{1,2,3,5},{4}}
 => {{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => 1000 => 1
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => 0100 => 0
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => [[1,4,5],[2],[3]]
 => 1100 => 2
{{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => 0100 => 0
{{1,2,4},{3,5}}
 => {{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => 1010 => 1
{{1,2,4},{3},{5}}
 => {{1},{2,4,5},{3}}
 => [[1,4,5],[2],[3]]
 => 1100 => 2
{{1,2,5},{3,4}}
 => {{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => 1010 => 1
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => 0010 => 0
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => [[1,3],[2,5],[4]]
 => 1010 => 1
{{1,2,5},{3},{4}}
 => {{1,4,5},{2},{3}}
 => [[1,4,5],[2],[3]]
 => 1100 => 2
{{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => 1010 => 1
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => 0110 => 0
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => [[1,5],[2],[3],[4]]
 => 1110 => 3
{{1,3,4,5},{2}}
 => {{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => 0010 => 0
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => 1010 => 1
{{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => [[1,3,5],[2],[4]]
 => 1010 => 1
{{1,3,5},{2,4}}
 => {{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => 1010 => 1
{{1,3},{2,4,5}}
 => {{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => 0101 => 0
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => [[1,4],[2,5],[3]]
 => 1101 => 2
{{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => 1010 => 1
{{1,3},{2,5},{4}}
 => {{1,4},{2},{3,5}}
 => [[1,4],[2,5],[3]]
 => 1101 => 2
{{1,3},{2},{4,5}}
 => {{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => 0110 => 0
{{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => [[1,5],[2],[3],[4]]
 => 1110 => 3
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => 0100 => 0
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => 1001 => 1
{{1,4},{2,3},{5}}
 => {{1},{2,5},{3,4}}
 => [[1,4],[2,5],[3]]
 => 1101 => 2
Description
The number of leading ones in a binary word.
Matching statistic: St000989
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [1,3,2] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [3,1,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 2
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [1,4,2,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [1,3,4,2] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [3,4,1,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [4,2,1,3] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [2,1,4,3] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,5,4] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [1,2,4,3,5] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [1,2,5,3,4] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [2,4,3,5,1] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [1,2,4,5,3] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [1,4,2,3,5] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [1,5,3,2,4] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [3,2,1,5,4] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [3,2,4,1,5] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [1,5,2,3,4] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [3,2,5,1,4] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,5,1] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [1,3,2,5,4] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [2,5,3,1,4] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [1,4,5,2,3] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [3,1,4,5,2] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [3,4,1,2,5] => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [3,5,1,2,4] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [4,2,3,5,1] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 3
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [1,4,3,5,2] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [3,1,5,2,4] => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [3,2,4,1,5] => [4,2,1,3,5] => 2
Description
The number of final rises of a permutation. For a permutation π of length n, this is the maximal k such that π(n−k)≤π(n−k+1)≤⋯≤π(n−1)≤π(n). Equivalently, this is n−1 minus the position of the last descent [[St000653]].
Matching statistic: St000678
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => [1,1,0,0]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
Description
The number of up steps after the last double rise of a Dyck path.