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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001050
(load all 100 compositions to match this statistic)
(load all 100 compositions to match this statistic)
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 1
{{1,2}}
=> 1
{{1},{2}}
=> 2
{{1,2,3}}
=> 1
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 1
{{1,2,3},{4}}
=> 2
{{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> 3
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 3
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 2
{{1,4},{2},{3}}
=> 3
{{1},{2,4},{3}}
=> 2
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 4
{{1,2,3,4,5}}
=> 1
{{1,2,3,4},{5}}
=> 2
{{1,2,3,5},{4}}
=> 2
{{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> 3
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 3
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 1
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 3
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 1
{{1,2},{3},{4},{5}}
=> 4
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 2
{{1,3,4},{2},{5}}
=> 2
{{1,3,5},{2,4}}
=> 2
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 3
{{1,3,5},{2},{4}}
=> 2
{{1,3},{2,5},{4}}
=> 3
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 4
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 2
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 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00112: Set partitions —complement⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> [[1]]
=> 1
{{1,2}}
=> {{1,2}}
=> [[1,2]]
=> 1
{{1},{2}}
=> {{1},{2}}
=> [[1],[2]]
=> 2
{{1,2,3}}
=> {{1,2,3}}
=> [[1,2,3]]
=> 1
{{1,2},{3}}
=> {{1},{2,3}}
=> [[1,3],[2]]
=> 2
{{1,3},{2}}
=> {{1,3},{2}}
=> [[1,3],[2]]
=> 2
{{1},{2,3}}
=> {{1,2},{3}}
=> [[1,2],[3]]
=> 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [[1],[2],[3]]
=> 3
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [[1,2,3,4]]
=> 1
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [[1,3,4],[2]]
=> 2
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [[1,3,4],[2]]
=> 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [[1,2],[3,4]]
=> 1
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> 3
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [[1,2,4],[3]]
=> 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [[1,3],[2,4]]
=> 2
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> 3
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [[1,3],[2,4]]
=> 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [[1,2,3],[4]]
=> 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> 2
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> 3
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> 2
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> 4
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> 1
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> [[1,3,4,5],[2]]
=> 2
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> 2
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> 1
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> [[1,4,5],[2],[3]]
=> 3
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> 2
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> [[1,4,5],[2],[3]]
=> 3
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> 2
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> 1
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> [[1,3],[2,5],[4]]
=> 2
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> [[1,4,5],[2],[3]]
=> 3
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> 2
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> 1
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> [[1,5],[2],[3],[4]]
=> 4
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> 2
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [[1,3,5],[2],[4]]
=> 2
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> 2
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> 1
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> [[1,4],[2,5],[3]]
=> 3
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> 2
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> [[1,4],[2,5],[3]]
=> 3
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> [[1,5],[2],[3],[4]]
=> 4
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> 2
{{1,2},{3,4},{5,6},{7,8},{9,10}}
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> ?
=> ? = 1
{{1,4},{2,3},{5,6},{7,8},{9,10}}
=> {{1,2},{3,4},{5,6},{7,10},{8,9}}
=> ?
=> ? = 1
{{1,6},{2,3},{4,5},{7,8},{9,10}}
=> {{1,2},{3,4},{5,10},{6,7},{8,9}}
=> ?
=> ? = 1
{{1,8},{2,3},{4,5},{6,7},{9,10}}
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> ?
=> ? = 1
{{1,10},{2,3},{4,5},{6,7},{8,9}}
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> ?
=> ? = 2
{{1,2},{3,6},{4,5},{7,8},{9,10}}
=> {{1,2},{3,4},{5,8},{6,7},{9,10}}
=> ?
=> ? = 1
{{1,6},{2,5},{3,4},{7,8},{9,10}}
=> {{1,2},{3,4},{5,10},{6,9},{7,8}}
=> ?
=> ? = 1
{{1,8},{2,5},{3,4},{6,7},{9,10}}
=> {{1,2},{3,10},{4,5},{6,9},{7,8}}
=> ?
=> ? = 1
{{1,10},{2,5},{3,4},{6,7},{8,9}}
=> {{1,10},{2,3},{4,5},{6,9},{7,8}}
=> ?
=> ? = 2
{{1,2},{3,8},{4,5},{6,7},{9,10}}
=> {{1,2},{3,8},{4,5},{6,7},{9,10}}
=> ?
=> ? = 1
{{1,8},{2,7},{3,4},{5,6},{9,10}}
=> {{1,2},{3,10},{4,9},{5,6},{7,8}}
=> ?
=> ? = 1
{{1,10},{2,7},{3,4},{5,6},{8,9}}
=> {{1,10},{2,3},{4,9},{5,6},{7,8}}
=> ?
=> ? = 2
{{1,2},{3,10},{4,5},{6,7},{8,9}}
=> {{1,8},{2,3},{4,5},{6,7},{9,10}}
=> ?
=> ? = 2
{{1,10},{2,9},{3,4},{5,6},{7,8}}
=> {{1,10},{2,9},{3,4},{5,6},{7,8}}
=> ?
=> ? = 3
{{1,2},{3,4},{5,8},{6,7},{9,10}}
=> {{1,2},{3,6},{4,5},{7,8},{9,10}}
=> ?
=> ? = 1
{{1,4},{2,3},{5,8},{6,7},{9,10}}
=> {{1,2},{3,6},{4,5},{7,10},{8,9}}
=> ?
=> ? = 1
{{1,8},{2,3},{4,7},{5,6},{9,10}}
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> ?
=> ? = 1
{{1,10},{2,3},{4,7},{5,6},{8,9}}
=> {{1,10},{2,3},{4,7},{5,6},{8,9}}
=> ?
=> ? = 2
{{1,2},{3,8},{4,7},{5,6},{9,10}}
=> {{1,2},{3,8},{4,7},{5,6},{9,10}}
=> ?
=> ? = 1
{{1,8},{2,7},{3,6},{4,5},{9,10}}
=> {{1,2},{3,10},{4,9},{5,8},{6,7}}
=> ?
=> ? = 1
{{1,10},{2,7},{3,6},{4,5},{8,9}}
=> {{1,10},{2,3},{4,9},{5,8},{6,7}}
=> ?
=> ? = 2
{{1,2},{3,10},{4,7},{5,6},{8,9}}
=> {{1,8},{2,3},{4,7},{5,6},{9,10}}
=> ?
=> ? = 2
{{1,10},{2,9},{3,6},{4,5},{7,8}}
=> {{1,10},{2,9},{3,4},{5,8},{6,7}}
=> ?
=> ? = 3
{{1,2},{3,4},{5,10},{6,7},{8,9}}
=> {{1,6},{2,3},{4,5},{7,8},{9,10}}
=> ?
=> ? = 2
{{1,4},{2,3},{5,10},{6,7},{8,9}}
=> {{1,6},{2,3},{4,5},{7,10},{8,9}}
=> ?
=> ? = 2
{{1,10},{2,3},{4,9},{5,6},{7,8}}
=> {{1,10},{2,7},{3,4},{5,6},{8,9}}
=> ?
=> ? = 3
{{1,2},{3,10},{4,9},{5,6},{7,8}}
=> {{1,8},{2,7},{3,4},{5,6},{9,10}}
=> ?
=> ? = 3
{{1,10},{2,9},{3,8},{4,5},{6,7}}
=> {{1,10},{2,9},{3,8},{4,5},{6,7}}
=> ?
=> ? = 4
{{1,2},{3,4},{5,6},{7,10},{8,9}}
=> {{1,4},{2,3},{5,6},{7,8},{9,10}}
=> ?
=> ? = 2
{{1,4},{2,3},{5,6},{7,10},{8,9}}
=> {{1,4},{2,3},{5,6},{7,10},{8,9}}
=> ?
=> ? = 2
{{1,6},{2,3},{4,5},{7,10},{8,9}}
=> {{1,4},{2,3},{5,10},{6,7},{8,9}}
=> ?
=> ? = 2
{{1,10},{2,3},{4,5},{6,9},{7,8}}
=> {{1,10},{2,5},{3,4},{6,7},{8,9}}
=> ?
=> ? = 3
{{1,2},{3,6},{4,5},{7,10},{8,9}}
=> {{1,4},{2,3},{5,8},{6,7},{9,10}}
=> ?
=> ? = 2
{{1,6},{2,5},{3,4},{7,10},{8,9}}
=> {{1,4},{2,3},{5,10},{6,9},{7,8}}
=> ?
=> ? = 2
{{1,10},{2,5},{3,4},{6,9},{7,8}}
=> {{1,10},{2,5},{3,4},{6,9},{7,8}}
=> ?
=> ? = 3
{{1,2},{3,10},{4,5},{6,9},{7,8}}
=> {{1,8},{2,5},{3,4},{6,7},{9,10}}
=> ?
=> ? = 3
{{1,10},{2,9},{3,4},{5,8},{6,7}}
=> {{1,10},{2,9},{3,6},{4,5},{7,8}}
=> ?
=> ? = 4
{{1,2},{3,4},{5,10},{6,9},{7,8}}
=> {{1,6},{2,5},{3,4},{7,8},{9,10}}
=> ?
=> ? = 3
{{1,4},{2,3},{5,10},{6,9},{7,8}}
=> {{1,6},{2,5},{3,4},{7,10},{8,9}}
=> ?
=> ? = 3
{{1,10},{2,3},{4,9},{5,8},{6,7}}
=> {{1,10},{2,7},{3,6},{4,5},{8,9}}
=> ?
=> ? = 4
{{1,2},{3,10},{4,9},{5,8},{6,7}}
=> {{1,8},{2,7},{3,6},{4,5},{9,10}}
=> ?
=> ? = 4
{{1,10},{2,9},{3,8},{4,7},{5,6}}
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> ?
=> ? = 5
{{1},{2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ?
=> ? = 1
{{1},{2,4,5,6,7,8},{3}}
=> {{1,2,3,4,5,7},{6},{8}}
=> ?
=> ? = 1
{{1},{2,3,5,6,7,8},{4}}
=> {{1,2,3,4,6,7},{5},{8}}
=> ?
=> ? = 1
{{1},{2,3,4,6,7,8},{5}}
=> {{1,2,3,5,6,7},{4},{8}}
=> ?
=> ? = 1
{{1},{2,3,4,5,8},{6},{7}}
=> {{1,4,5,6,7},{2},{3},{8}}
=> ?
=> ? = 3
{{1},{2,3,4,5,7,8},{6}}
=> {{1,2,4,5,6,7},{3},{8}}
=> ?
=> ? = 1
{{1},{2,3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6,7},{8}}
=> ?
=> ? = 2
{{1},{2,3,4,8},{5,6,7}}
=> {{1,5,6,7},{2,3,4},{8}}
=> ?
=> ? = 2
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)
(load all 2 compositions to match this statistic)
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: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> [[1]]
=> => ? = 1 - 1
{{1,2}}
=> {{1,2}}
=> [[1,2]]
=> 0 => 0 = 1 - 1
{{1},{2}}
=> {{1},{2}}
=> [[1],[2]]
=> 1 => 1 = 2 - 1
{{1,2,3}}
=> {{1,2,3}}
=> [[1,2,3]]
=> 00 => 0 = 1 - 1
{{1,2},{3}}
=> {{1},{2,3}}
=> [[1,3],[2]]
=> 10 => 1 = 2 - 1
{{1,3},{2}}
=> {{1,3},{2}}
=> [[1,3],[2]]
=> 10 => 1 = 2 - 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [[1,2],[3]]
=> 01 => 0 = 1 - 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [[1],[2],[3]]
=> 11 => 2 = 3 - 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [[1,2,3,4]]
=> 000 => 0 = 1 - 1
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [[1,3,4],[2]]
=> 100 => 1 = 2 - 1
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [[1,3,4],[2]]
=> 100 => 1 = 2 - 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [[1,2],[3,4]]
=> 010 => 0 = 1 - 1
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> 110 => 2 = 3 - 1
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [[1,2,4],[3]]
=> 010 => 0 = 1 - 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [[1,3],[2,4]]
=> 101 => 1 = 2 - 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> 110 => 2 = 3 - 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [[1,3],[2,4]]
=> 101 => 1 = 2 - 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [[1,2,3],[4]]
=> 001 => 0 = 1 - 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> 101 => 1 = 2 - 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> 110 => 2 = 3 - 1
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> 101 => 1 = 2 - 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> 011 => 0 = 1 - 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> 111 => 3 = 4 - 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> 0000 => 0 = 1 - 1
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> [[1,3,4,5],[2]]
=> 1000 => 1 = 2 - 1
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> 1000 => 1 = 2 - 1
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> 0100 => 0 = 1 - 1
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 3 - 1
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> 0100 => 0 = 1 - 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> 1010 => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 3 - 1
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> 1010 => 1 = 2 - 1
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> 0010 => 0 = 1 - 1
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> [[1,3],[2,5],[4]]
=> 1010 => 1 = 2 - 1
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 3 - 1
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> 1010 => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> 0110 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 4 - 1
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> 0010 => 0 = 1 - 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> 1010 => 1 = 2 - 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [[1,3,5],[2],[4]]
=> 1010 => 1 = 2 - 1
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> 1010 => 1 = 2 - 1
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> 0101 => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> [[1,4],[2,5],[3]]
=> 1101 => 2 = 3 - 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> 1010 => 1 = 2 - 1
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> [[1,4],[2,5],[3]]
=> 1101 => 2 = 3 - 1
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> 0110 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 4 - 1
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> 0100 => 0 = 1 - 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> 1001 => 1 = 2 - 1
{{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> [[1,4],[2,5],[3]]
=> 1101 => 2 = 3 - 1
{{1,2},{3,4},{5,6},{7,8},{9,10}}
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> ?
=> ? => ? = 1 - 1
{{1,4},{2,3},{5,6},{7,8},{9,10}}
=> {{1,2},{3,4},{5,6},{7,10},{8,9}}
=> ?
=> ? => ? = 1 - 1
{{1,6},{2,3},{4,5},{7,8},{9,10}}
=> {{1,2},{3,4},{5,10},{6,7},{8,9}}
=> ?
=> ? => ? = 1 - 1
{{1,8},{2,3},{4,5},{6,7},{9,10}}
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> ?
=> ? => ? = 1 - 1
{{1,10},{2,3},{4,5},{6,7},{8,9}}
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> ?
=> ? => ? = 2 - 1
{{1,2},{3,6},{4,5},{7,8},{9,10}}
=> {{1,2},{3,4},{5,8},{6,7},{9,10}}
=> ?
=> ? => ? = 1 - 1
{{1,6},{2,5},{3,4},{7,8},{9,10}}
=> {{1,2},{3,4},{5,10},{6,9},{7,8}}
=> ?
=> ? => ? = 1 - 1
{{1,8},{2,5},{3,4},{6,7},{9,10}}
=> {{1,2},{3,10},{4,5},{6,9},{7,8}}
=> ?
=> ? => ? = 1 - 1
{{1,10},{2,5},{3,4},{6,7},{8,9}}
=> {{1,10},{2,3},{4,5},{6,9},{7,8}}
=> ?
=> ? => ? = 2 - 1
{{1,2},{3,8},{4,5},{6,7},{9,10}}
=> {{1,2},{3,8},{4,5},{6,7},{9,10}}
=> ?
=> ? => ? = 1 - 1
{{1,8},{2,7},{3,4},{5,6},{9,10}}
=> {{1,2},{3,10},{4,9},{5,6},{7,8}}
=> ?
=> ? => ? = 1 - 1
{{1,10},{2,7},{3,4},{5,6},{8,9}}
=> {{1,10},{2,3},{4,9},{5,6},{7,8}}
=> ?
=> ? => ? = 2 - 1
{{1,2},{3,10},{4,5},{6,7},{8,9}}
=> {{1,8},{2,3},{4,5},{6,7},{9,10}}
=> ?
=> ? => ? = 2 - 1
{{1,10},{2,9},{3,4},{5,6},{7,8}}
=> {{1,10},{2,9},{3,4},{5,6},{7,8}}
=> ?
=> ? => ? = 3 - 1
{{1,2},{3,4},{5,8},{6,7},{9,10}}
=> {{1,2},{3,6},{4,5},{7,8},{9,10}}
=> ?
=> ? => ? = 1 - 1
{{1,4},{2,3},{5,8},{6,7},{9,10}}
=> {{1,2},{3,6},{4,5},{7,10},{8,9}}
=> ?
=> ? => ? = 1 - 1
{{1,8},{2,3},{4,7},{5,6},{9,10}}
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> ?
=> ? => ? = 1 - 1
{{1,10},{2,3},{4,7},{5,6},{8,9}}
=> {{1,10},{2,3},{4,7},{5,6},{8,9}}
=> ?
=> ? => ? = 2 - 1
{{1,2},{3,8},{4,7},{5,6},{9,10}}
=> {{1,2},{3,8},{4,7},{5,6},{9,10}}
=> ?
=> ? => ? = 1 - 1
{{1,8},{2,7},{3,6},{4,5},{9,10}}
=> {{1,2},{3,10},{4,9},{5,8},{6,7}}
=> ?
=> ? => ? = 1 - 1
{{1,10},{2,7},{3,6},{4,5},{8,9}}
=> {{1,10},{2,3},{4,9},{5,8},{6,7}}
=> ?
=> ? => ? = 2 - 1
{{1,2},{3,10},{4,7},{5,6},{8,9}}
=> {{1,8},{2,3},{4,7},{5,6},{9,10}}
=> ?
=> ? => ? = 2 - 1
{{1,10},{2,9},{3,6},{4,5},{7,8}}
=> {{1,10},{2,9},{3,4},{5,8},{6,7}}
=> ?
=> ? => ? = 3 - 1
{{1,2},{3,4},{5,10},{6,7},{8,9}}
=> {{1,6},{2,3},{4,5},{7,8},{9,10}}
=> ?
=> ? => ? = 2 - 1
{{1,4},{2,3},{5,10},{6,7},{8,9}}
=> {{1,6},{2,3},{4,5},{7,10},{8,9}}
=> ?
=> ? => ? = 2 - 1
{{1,10},{2,3},{4,9},{5,6},{7,8}}
=> {{1,10},{2,7},{3,4},{5,6},{8,9}}
=> ?
=> ? => ? = 3 - 1
{{1,2},{3,10},{4,9},{5,6},{7,8}}
=> {{1,8},{2,7},{3,4},{5,6},{9,10}}
=> ?
=> ? => ? = 3 - 1
{{1,10},{2,9},{3,8},{4,5},{6,7}}
=> {{1,10},{2,9},{3,8},{4,5},{6,7}}
=> ?
=> ? => ? = 4 - 1
{{1,2},{3,4},{5,6},{7,10},{8,9}}
=> {{1,4},{2,3},{5,6},{7,8},{9,10}}
=> ?
=> ? => ? = 2 - 1
{{1,4},{2,3},{5,6},{7,10},{8,9}}
=> {{1,4},{2,3},{5,6},{7,10},{8,9}}
=> ?
=> ? => ? = 2 - 1
{{1,6},{2,3},{4,5},{7,10},{8,9}}
=> {{1,4},{2,3},{5,10},{6,7},{8,9}}
=> ?
=> ? => ? = 2 - 1
{{1,10},{2,3},{4,5},{6,9},{7,8}}
=> {{1,10},{2,5},{3,4},{6,7},{8,9}}
=> ?
=> ? => ? = 3 - 1
{{1,2},{3,6},{4,5},{7,10},{8,9}}
=> {{1,4},{2,3},{5,8},{6,7},{9,10}}
=> ?
=> ? => ? = 2 - 1
{{1,6},{2,5},{3,4},{7,10},{8,9}}
=> {{1,4},{2,3},{5,10},{6,9},{7,8}}
=> ?
=> ? => ? = 2 - 1
{{1,10},{2,5},{3,4},{6,9},{7,8}}
=> {{1,10},{2,5},{3,4},{6,9},{7,8}}
=> ?
=> ? => ? = 3 - 1
{{1,2},{3,10},{4,5},{6,9},{7,8}}
=> {{1,8},{2,5},{3,4},{6,7},{9,10}}
=> ?
=> ? => ? = 3 - 1
{{1,10},{2,9},{3,4},{5,8},{6,7}}
=> {{1,10},{2,9},{3,6},{4,5},{7,8}}
=> ?
=> ? => ? = 4 - 1
{{1,2},{3,4},{5,10},{6,9},{7,8}}
=> {{1,6},{2,5},{3,4},{7,8},{9,10}}
=> ?
=> ? => ? = 3 - 1
{{1,4},{2,3},{5,10},{6,9},{7,8}}
=> {{1,6},{2,5},{3,4},{7,10},{8,9}}
=> ?
=> ? => ? = 3 - 1
{{1,10},{2,3},{4,9},{5,8},{6,7}}
=> {{1,10},{2,7},{3,6},{4,5},{8,9}}
=> ?
=> ? => ? = 4 - 1
{{1,2},{3,10},{4,9},{5,8},{6,7}}
=> {{1,8},{2,7},{3,6},{4,5},{9,10}}
=> ?
=> ? => ? = 4 - 1
{{1,10},{2,9},{3,8},{4,7},{5,6}}
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> ?
=> ? => ? = 5 - 1
{{1},{2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ?
=> ? => ? = 1 - 1
{{1},{2,4,5,6,7,8},{3}}
=> {{1,2,3,4,5,7},{6},{8}}
=> ?
=> ? => ? = 1 - 1
{{1},{2,3,5,6,7,8},{4}}
=> {{1,2,3,4,6,7},{5},{8}}
=> ?
=> ? => ? = 1 - 1
{{1},{2,3,4,6,7,8},{5}}
=> {{1,2,3,5,6,7},{4},{8}}
=> ?
=> ? => ? = 1 - 1
{{1},{2,3,4,5,8},{6},{7}}
=> {{1,4,5,6,7},{2},{3},{8}}
=> ?
=> ? => ? = 3 - 1
{{1},{2,3,4,5,7,8},{6}}
=> {{1,2,4,5,6,7},{3},{8}}
=> ?
=> ? => ? = 1 - 1
{{1},{2,3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6,7},{8}}
=> ?
=> ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000678
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: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> ? = 1
{{1,2}}
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
{{1},{2}}
=> [1,2] => [1,2] => [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
{{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
{{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
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{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
{{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
{{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
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
{{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
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{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
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
{{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
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{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
{{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
{{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
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
{{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
{{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
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
{{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
{{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
{{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
{{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
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
{{1,3},{2,4},{5,6},{7,8}}
=> [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,6},{2,3},{4,5},{7,8}}
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,7},{2,3},{4,5},{6,8}}
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,8},{2,3},{4,5},{6,7}}
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,8},{2,4},{3,5},{6,7}}
=> [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,7},{2,4},{3,5},{6,8}}
=> [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,6},{2,4},{3,5},{7,8}}
=> [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,4},{3,6},{7,8}}
=> [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,4},{2,5},{3,6},{7,8}}
=> [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,5},{4,6},{7,8}}
=> [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,4},{2,6},{3,5},{7,8}}
=> [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,6},{3,4},{7,8}}
=> [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,7},{2,5},{3,4},{6,8}}
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,8},{2,5},{3,4},{6,7}}
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,7},{2,6},{3,4},{5,8}}
=> [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,6},{2,7},{3,4},{5,8}}
=> [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,5},{2,7},{3,4},{6,8}}
=> [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,4},{2,7},{3,5},{6,8}}
=> [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,3},{2,7},{4,5},{6,8}}
=> [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,3},{2,8},{4,5},{6,7}}
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,4},{2,8},{3,5},{6,7}}
=> [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,5},{2,8},{3,4},{6,7}}
=> [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
{{1,6},{2,8},{3,4},{5,7}}
=> [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,7},{2,8},{3,4},{5,6}}
=> [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,8},{2,7},{3,5},{4,6}}
=> [8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,7},{2,8},{3,5},{4,6}}
=> [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,6},{2,8},{3,5},{4,7}}
=> [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,5},{2,8},{3,6},{4,7}}
=> [5,8,6,7,1,3,4,2] => [5,1,6,3,7,4,8,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,4},{2,8},{3,6},{5,7}}
=> [4,8,6,1,7,3,5,2] => [4,1,6,3,7,5,8,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,3},{2,8},{4,6},{5,7}}
=> [3,8,1,6,7,4,5,2] => [3,1,6,4,7,5,8,2] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,3},{2,7},{4,6},{5,8}}
=> [3,7,1,6,8,4,2,5] => [3,1,6,4,7,2,8,5] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,4},{2,7},{3,6},{5,8}}
=> [4,7,6,1,8,3,2,5] => [4,1,6,3,7,2,8,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,5},{2,7},{3,6},{4,8}}
=> [5,7,6,8,1,3,2,4] => [5,1,6,3,7,2,8,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,6},{2,7},{3,5},{4,8}}
=> [6,7,5,8,3,1,2,4] => [5,3,6,1,7,2,8,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,7},{2,6},{3,5},{4,8}}
=> [7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,8},{2,6},{3,5},{4,7}}
=> [8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,8},{2,5},{3,6},{4,7}}
=> [8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,7},{2,5},{3,6},{4,8}}
=> [7,5,6,8,2,3,1,4] => [5,2,6,3,7,1,8,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,6},{2,5},{3,7},{4,8}}
=> [6,5,7,8,2,1,3,4] => [5,2,6,1,7,3,8,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,5},{2,6},{3,7},{4,8}}
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,4},{2,6},{3,7},{5,8}}
=> [4,6,7,1,8,2,3,5] => [4,1,6,2,7,3,8,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,3},{2,6},{4,7},{5,8}}
=> [3,6,1,7,8,2,4,5] => [3,1,6,2,7,4,8,5] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,3},{2,5},{4,7},{6,8}}
=> [3,5,1,7,2,8,4,6] => [3,1,5,2,7,4,8,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000989
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => ? = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [1,3,2] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => [1,4,2,3] => 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => [1,3,4,2] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [3,4,1,2] => 1 = 2 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [4,2,1,3] => 1 = 2 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [2,1,4,3] => 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 1 = 2 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => [1,2,5,3,4] => 1 = 2 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [2,4,3,5,1] => 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => [1,2,4,5,3] => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => [2,4,5,1,3] => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => [1,4,2,3,5] => 2 = 3 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => [1,5,3,2,4] => 1 = 2 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [3,2,1,5,4] => 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,2,4,1,5] => 1 = 2 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => [1,5,2,3,4] => 2 = 3 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [3,2,5,1,4] => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,5,1] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => [1,3,2,5,4] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => [2,5,3,1,4] => 1 = 2 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => [1,3,4,2,5] => 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => [1,4,5,2,3] => 1 = 2 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => [3,1,4,5,2] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,4,1,2,5] => 2 = 3 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => [1,3,5,2,4] => 1 = 2 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => [3,5,1,2,4] => 2 = 3 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [4,2,3,5,1] => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 3 = 4 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => [1,4,3,5,2] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => [3,1,5,2,4] => 1 = 2 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => [4,2,1,3,5] => 2 = 3 - 1
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [5,1,2,3,4,7,6] => [2,3,4,6,5,7,1] => ? = 1 - 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [6,1,2,3,4,7,5] => [2,3,4,6,7,1,5] => ? = 2 - 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => [2,3,5,4,1,7,6] => ? = 1 - 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => [2,3,5,4,6,1,7] => ? = 2 - 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [4,1,2,3,6,7,5] => [2,3,5,4,7,1,6] => ? = 2 - 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => [2,3,5,4,6,7,1] => ? = 1 - 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [6,1,2,3,5,7,4] => [2,3,4,7,5,1,6] => ? = 2 - 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [5,1,2,3,7,4,6] => [2,3,5,1,6,7,4] => ? = 1 - 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [5,1,2,3,6,4,7] => [2,3,5,6,1,4,7] => ? = 3 - 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [5,1,2,3,6,7,4] => [2,3,5,7,1,4,6] => ? = 3 - 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [4,5,1,2,3,7,6] => [2,3,6,4,5,7,1] => ? = 1 - 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [6,1,2,3,7,4,5] => [2,3,5,1,7,4,6] => ? = 2 - 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => [2,4,3,1,5,7,6] => ? = 1 - 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [3,1,2,6,4,5,7] => [2,4,3,1,6,5,7] => ? = 2 - 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [3,1,2,6,7,4,5] => [2,4,3,1,7,5,6] => ? = 2 - 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => [3,5,4,6,2,7,1] => ? = 1 - 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [3,1,2,5,4,6,7] => [2,4,3,5,1,6,7] => ? = 3 - 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [5,6,1,2,3,7,4] => [2,3,6,7,1,4,5] => ? = 3 - 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [4,6,1,2,3,7,5] => [2,3,6,4,7,1,5] => ? = 2 - 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [3,1,2,5,7,4,6] => [2,4,3,1,6,7,5] => ? = 1 - 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [3,1,2,6,4,7,5] => [3,5,4,6,7,1,2] => ? = 2 - 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [3,1,2,5,6,4,7] => [2,4,3,6,1,5,7] => ? = 3 - 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [3,1,2,6,5,7,4] => [3,5,4,7,2,1,6] => ? = 2 - 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => [2,4,3,5,1,7,6] => ? = 1 - 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [3,1,2,4,6,5,7] => [2,4,3,5,6,1,7] => ? = 2 - 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [3,1,2,5,6,7,4] => [2,4,3,7,1,5,6] => ? = 3 - 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [3,1,2,4,6,7,5] => [2,4,3,5,7,1,6] => ? = 2 - 1
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 1 - 1
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [6,1,2,4,5,7,3] => [2,3,7,1,5,4,6] => ? = 2 - 1
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => [5,1,2,4,7,3,6] => [2,3,1,6,5,7,4] => ? = 1 - 1
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => [5,1,2,4,6,3,7] => [2,3,6,4,1,5,7] => ? = 3 - 1
{{1,2,4,5},{3,7},{6}}
=> [2,4,7,5,1,6,3] => [5,1,2,4,6,7,3] => [2,3,7,4,1,5,6] => ? = 3 - 1
{{1,2,4,5},{3},{6,7}}
=> [2,4,3,5,1,7,6] => [3,5,1,2,4,7,6] => [2,3,5,6,4,7,1] => ? = 1 - 1
{{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => [6,1,2,4,7,3,5] => [2,3,1,6,7,4,5] => ? = 2 - 1
{{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => [4,1,2,7,3,5,6] => [2,4,1,5,3,7,6] => ? = 1 - 1
{{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => [4,1,2,6,3,5,7] => [2,4,1,5,6,3,7] => ? = 2 - 1
{{1,2,4},{3,5,7},{6}}
=> [2,4,5,1,7,6,3] => [4,1,2,6,7,3,5] => [2,4,1,5,7,3,6] => ? = 2 - 1
{{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => [4,1,2,5,3,7,6] => [3,5,6,2,4,7,1] => ? = 1 - 1
{{1,2,4},{3,5},{6},{7}}
=> [2,4,5,1,3,6,7] => [4,1,2,5,3,6,7] => [2,4,5,1,3,6,7] => ? = 4 - 1
{{1,2,4,6},{3,7},{5}}
=> [2,4,7,6,5,1,3] => [5,6,1,2,4,7,3] => [2,3,7,5,1,4,6] => ? = 3 - 1
{{1,2,4,6},{3},{5,7}}
=> [2,4,3,6,7,1,5] => [3,6,1,2,4,7,5] => [2,3,5,6,7,1,4] => ? = 2 - 1
{{1,2,4},{3,6,7},{5}}
=> [2,4,6,1,5,7,3] => [4,1,2,5,7,3,6] => [2,4,1,5,6,7,3] => ? = 1 - 1
{{1,2,4},{3,6},{5,7}}
=> [2,4,6,1,7,3,5] => [4,1,2,6,3,7,5] => [3,5,6,2,7,1,4] => ? = 2 - 1
{{1,2,4},{3,6},{5},{7}}
=> [2,4,6,1,5,3,7] => [4,1,2,5,6,3,7] => [2,4,6,1,3,5,7] => ? = 4 - 1
{{1,2,4},{3,7},{5,6}}
=> [2,4,7,1,6,5,3] => [4,1,2,6,5,7,3] => [3,5,7,2,4,1,6] => ? = 2 - 1
{{1,2,4},{3},{5,6,7}}
=> [2,4,3,1,6,7,5] => [3,4,1,2,7,5,6] => [2,5,3,4,1,7,6] => ? = 1 - 1
{{1,2,4},{3},{5,6},{7}}
=> [2,4,3,1,6,5,7] => [3,4,1,2,6,5,7] => [2,5,3,4,6,1,7] => ? = 2 - 1
{{1,2,4},{3,7},{5},{6}}
=> [2,4,7,1,5,6,3] => [4,1,2,5,6,7,3] => [2,4,7,1,3,5,6] => ? = 4 - 1
{{1,2,4},{3},{5,7},{6}}
=> [2,4,3,1,7,6,5] => [3,4,1,2,6,7,5] => [2,5,3,4,7,1,6] => ? = 2 - 1
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
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