Your data matches 49 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001211
(load all 23 compositions to match this statistic)
Mapping
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 2
[1,0,1,0]
 => 2
[1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => 3
[1,1,0,1,0,0]
 => 3
[1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,1,0,0,0]
 => 4
[1,1,1,0,0,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => 4
[1,1,1,0,1,0,0,0]
 => 4
[1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => 5
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
(load all 6 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => 2
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 2
[1,1,0,0]
 => [2] => [1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 3
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 3
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 5
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000507
(load all 5 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [[1]]
 => 1 = 2 - 1
[1,0,1,0]
 => [2,1] => [[1],[2]]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [[1,2]]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [[1,3],[2]]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [[1,3],[2]]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [[1,2],[3]]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [[1,2],[3]]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [[1,2,3]]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [[1,3,4],[2]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [[1,3],[2,4]]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [[1,2,4],[3]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [[1,2],[3,4]]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [[1,2,4],[3]]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [[1,2,3],[4]]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [[1,2,3],[4]]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [[1,2,4,5],[3]]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [[1,2,4,5],[3]]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [[1,2,4,5],[3]]
 => 4 = 5 - 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000393
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2] => 10 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => 101 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => 100 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 1011 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => 1010 => 3
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => 1001 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => 1001 => 3
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 1000 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => 10111 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => 10110 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => 10101 => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => 10101 => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => 10100 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => 10011 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => 10010 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => 10011 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => 10011 => 4
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => 10010 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => 10001 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => 10001 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => 10001 => 4
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => 10000 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => 101111 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => 101110 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => 101101 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => 101101 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => 101100 => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => 101011 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => 101010 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => 101011 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => 101011 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => 101010 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => 101001 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => 101001 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => 101001 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => 101000 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => 100111 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => 100110 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => 100101 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => 100101 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => 100100 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => 100111 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => 100110 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => 100111 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => 100111 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => 100110 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => 100101 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => 100101 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => 100101 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => 100100 => 5
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000010
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => [1]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 4 = 5 - 1
Description
The length of the partition.
Matching statistic: St000676
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 5 - 1
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001007
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4 = 5 - 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001068
(load all 2 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 5 - 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001250
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => [1]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 4 = 5 - 1
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St000024
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => 0 = 2 - 2
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 0 = 2 - 2
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 4 - 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 5 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 5 - 2
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000245The number of ascents of a permutation. St000702The number of weak deficiencies of a permutation. St000167The number of leaves of an ordered tree. St000672The number of minimal elements in Bruhat order not less than the permutation. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000308The height of the tree associated to a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001896The number of right descents of a signed permutations. St000331The number of upper interactions of a Dyck path. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001626The number of maximal proper sublattices of a lattice.