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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000740
(load all 42 compositions to match this statistic)
(load all 42 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 2
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 4
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000839
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => {{1}}
=> 1
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> 2
[1,1,0,0]
=> [2,1] => {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => {{1,3},{2}}
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 4
Description
The largest opener of a set partition.
An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.
Matching statistic: St000025
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,2] => [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 4
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000734
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 2
[1,1,0,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [[1,2],[3]]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [[1,2,4],[3],[5]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [[1,4],[2],[3],[5]]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [[1,3,5],[2],[4]]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [[1,3],[2],[4],[5]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [[1,3,4],[2],[5]]
=> 4
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000738
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [2,1] => [[1],[2]]
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => [[1,2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => [[1,2],[3]]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => [[1,3],[2]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => [[1],[2],[3]]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [[1,3],[2]]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => [[1,2,4],[3]]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => [[1,2],[3],[4]]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,3,4] => [[1,3,4],[2]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => [[1,3],[2],[4]]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,4,2,1] => [[1,2],[3],[4]]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,1,2,4] => [[1,3,4],[2]]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,2,3] => [[1,3,4],[2]]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,5,3,4,1] => [[1,2,4],[3],[5]]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,5,3,1,4] => [[1,2,5],[3],[4]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2,5,1,4] => [[1,3],[2,5],[4]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [[1,3,5],[2],[4]]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => [[1,3],[2,4],[5]]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,3,4] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> 4
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000439
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,2] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000476
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St001497
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,2,4,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [2,1,4,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [1,3,2,4] => 4
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => [2,4,1,3] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [1,2,4,3] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,3,5,4] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,3,2,5,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,2,4,3,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,2,4] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,2,3,5,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,3,5,4] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [3,1,2,5,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [2,1,4,3,5] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [3,1,5,2,4] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [3,1,5,2,4] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [2,1,3,5,4] => 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 7
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 6
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 7
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => [1,5,2,3,4,7,6] => ? = 6
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => [1,5,2,3,7,4,6] => ? = 5
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => [1,5,2,3,7,4,6] => ? = 5
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => [1,5,2,7,3,4,6] => ? = 4
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => [1,5,2,7,3,4,6] => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,2,6,7,3] => [1,4,7,2,3,5,6] => [1,4,7,2,3,5,6] => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,2,7,3] => [1,5,7,2,3,4,6] => [1,5,7,2,3,4,6] => ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => [1,5,7,2,3,4,6] => ? = 3
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St000653
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1 - 1
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => [1,3,2] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => [1,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => [2,1,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,2,3,1] => [4,1,3,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,4,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => [1,2,4,3,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => [1,2,5,4,3] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,1,4,5] => [1,3,2,4,5] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,3,5,1,4] => [1,3,5,2,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => [1,3,2,5,4] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => [1,4,3,2,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,5,3,4,1] => [1,5,2,4,3] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,5,3,1,4] => [1,5,3,2,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,4,5,3,1] => [1,2,5,4,3] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,4,1,3,5] => [2,4,1,3,5] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,5,1,3,4] => [2,5,1,3,4] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,4,5,1,3] => [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => [2,1,3,5,4] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => [2,1,4,3,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => [2,1,5,4,3] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,1,4,5] => [3,2,1,4,5] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2,5,1,4] => [3,2,5,1,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,2,3,5,1] => [3,1,2,5,4] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,1,5] => [4,1,3,2,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,2,3,4,1] => [5,1,2,4,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [5,1,3,2,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => [2,1,5,4,3] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => [4,2,1,3,5] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,3,4] => [5,2,1,3,4] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => [3,2,5,1,4] => 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,4,2,5,1] => [1,3,2,5,4] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [2,5,3,4,1,6,7] => [1,5,2,4,3,6,7] => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,5,6] => [2,5,3,4,7,1,6] => [1,5,2,4,7,3,6] => ? = 6 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [2,6,3,4,5,7,1] => [1,5,2,3,4,7,6] => ? = 7 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [2,6,3,4,5,1,7] => [1,6,2,3,5,4,7] => ? = 6 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [2,7,3,4,5,6,1] => [1,7,2,3,4,6,5] => ? = 7 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6] => [2,7,3,4,5,1,6] => [1,7,2,3,5,4,6] => ? = 6 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => [2,6,3,4,1,5,7] => [1,6,2,4,3,5,7] => ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,2,5] => [2,7,3,4,1,5,6] => [1,7,2,4,3,5,6] => ? = 5 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => [2,6,3,4,7,1,5] => [1,5,2,4,7,3,6] => ? = 6 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,2,6,7,4] => [2,5,3,1,4,6,7] => [1,5,3,2,4,6,7] => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,7,4,6] => [2,5,3,7,4,1,6] => [1,5,2,7,4,3,6] => ? = 6 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,2,7,4] => [2,6,3,1,4,5,7] => [1,6,3,2,4,5,7] => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,2,4] => [2,7,3,1,4,5,6] => [1,7,3,2,4,5,6] => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,2,4,6] => [2,6,3,7,4,1,5] => [1,5,2,7,4,3,6] => ? = 6 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,2,7,4,5] => [2,5,3,7,1,4,6] => [1,5,3,7,2,4,6] => ? = 5 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,4,5] => [2,6,3,7,1,4,5] => [1,5,3,7,2,4,6] => ? = 5 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => [2,4,1,3,5,6,7] => [2,4,1,3,5,6,7] => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,2,6,7,3,5] => [2,4,7,3,1,5,6] => [1,4,7,3,2,5,6] => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,4,5,2,6,3,7] => [2,5,7,3,4,6,1] => [1,4,7,2,3,6,5] => ? = 7 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,2,6,7,3] => [2,5,1,3,4,6,7] => [2,5,1,3,4,6,7] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,2,7,3,6] => [2,5,7,3,4,1,6] => [1,5,7,2,4,3,6] => ? = 6 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,2,3,7] => [2,6,7,3,4,5,1] => [1,4,7,2,3,6,5] => ? = 7 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,2,7,3] => [2,6,1,3,4,5,7] => [2,6,1,3,4,5,7] => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,2,3] => [2,7,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,2,3,6] => [2,6,7,3,4,1,5] => [1,5,7,2,4,3,6] => ? = 6 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,2,7,3,5] => [2,5,7,3,1,4,6] => [1,5,7,3,2,4,6] => ? = 5 - 1
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000147
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 5 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 7 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 7 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 7 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 7 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> ? = 6 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 7 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 7 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 6 - 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 7 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> ? = 6 - 1
Description
The largest part of an integer partition.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000727The largest label of a leaf in the binary search tree associated with the permutation. St000019The cardinality of the support of a permutation. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000067The inversion number of the alternating sign matrix. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000240The number of indices that are not small excedances. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000957The number of Bruhat lower covers of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000840The number of closers smaller than the largest opener in a perfect matching. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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