Your data matches 188 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000764
(load all 7 compositions to match this statistic)
Mapping
St000764: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 0 + 1
[1,1] => 1 = 0 + 1
[2] => 1 = 0 + 1
[1,1,1] => 1 = 0 + 1
[1,2] => 2 = 1 + 1
[2,1] => 1 = 0 + 1
[3] => 1 = 0 + 1
[1,1,1,1] => 1 = 0 + 1
[1,1,2] => 2 = 1 + 1
[1,2,1] => 2 = 1 + 1
[1,3] => 2 = 1 + 1
[2,1,1] => 1 = 0 + 1
[2,2] => 1 = 0 + 1
[3,1] => 1 = 0 + 1
[4] => 1 = 0 + 1
[1,1,1,1,1] => 1 = 0 + 1
[1,1,1,2] => 2 = 1 + 1
[1,1,2,1] => 2 = 1 + 1
[1,1,3] => 2 = 1 + 1
[1,2,1,1] => 2 = 1 + 1
[1,2,2] => 2 = 1 + 1
[1,3,1] => 2 = 1 + 1
[1,4] => 2 = 1 + 1
[2,1,1,1] => 1 = 0 + 1
[2,1,2] => 1 = 0 + 1
[2,2,1] => 1 = 0 + 1
[2,3] => 2 = 1 + 1
[3,1,1] => 1 = 0 + 1
[3,2] => 1 = 0 + 1
[4,1] => 1 = 0 + 1
[5] => 1 = 0 + 1
[1,1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,2] => 2 = 1 + 1
[1,1,1,2,1] => 2 = 1 + 1
[1,1,1,3] => 2 = 1 + 1
[1,1,2,1,1] => 2 = 1 + 1
[1,1,2,2] => 2 = 1 + 1
[1,1,3,1] => 2 = 1 + 1
[1,1,4] => 2 = 1 + 1
[1,2,1,1,1] => 2 = 1 + 1
[1,2,1,2] => 2 = 1 + 1
[1,2,2,1] => 2 = 1 + 1
[1,2,3] => 3 = 2 + 1
[1,3,1,1] => 2 = 1 + 1
[1,3,2] => 2 = 1 + 1
[1,4,1] => 2 = 1 + 1
[1,5] => 2 = 1 + 1
[2,1,1,1,1] => 1 = 0 + 1
[2,1,1,2] => 1 = 0 + 1
[2,1,2,1] => 1 = 0 + 1
Description
The number of strong records in an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Matching statistic: St001732
(load all 7 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 1 = 0 + 1
[1,1] => [1,0,1,0]
 => 1 = 0 + 1
[2] => [1,1,0,0]
 => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
Description
The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000660
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 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]
 => [1,0,1,1,0,0,1,1,0,0]
 => 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]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0
Description
The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001085
(load all 5 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => [1] => 0
[1,1] => [1,0,1,0]
 => [1,2] => [1,2] => 0
[2] => [1,1,0,0]
 => [2,1] => [2,1] => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,2] => [1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 0
[2,1] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[3] => [1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,4,2,3] => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 1
[4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,5,1,3,4] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,5,2,3,4] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,4,2,3,5] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,1,4,2,3] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,1,5,3,4] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [6,5,1,2,3,4] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [4,1,2,3,5,6] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [3,6,1,2,4,5] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [5,4,1,2,3,6] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [2,6,1,3,4,5] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [2,5,1,3,4,6] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => [6,2,5,1,3,4] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => [4,3,1,2,5,6] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => [3,2,6,1,4,5] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => [5,4,3,1,2,6] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [6,5,4,3,1,2] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => [1,6,2,3,4,5] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => [1,5,2,3,4,6] => 0
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001172
(load all 3 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,2] => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[3] => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 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]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 1
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001186
(load all 2 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 0
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 1
Description
Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St000506
(load all 7 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => []
 => []
 => ? = 0
[1,1] => [1,1]
 => [1]
 => [1]
 => 0
[2] => [2]
 => []
 => []
 => ? = 0
[1,1,1] => [1,1,1]
 => [1,1]
 => [2]
 => 0
[1,2] => [2,1]
 => [1]
 => [1]
 => 0
[2,1] => [2,1]
 => [1]
 => [1]
 => 0
[3] => [3]
 => []
 => []
 => ? = 1
[1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,1,2] => [2,1,1]
 => [1,1]
 => [2]
 => 0
[1,2,1] => [2,1,1]
 => [1,1]
 => [2]
 => 0
[1,3] => [3,1]
 => [1]
 => [1]
 => 0
[2,1,1] => [2,1,1]
 => [1,1]
 => [2]
 => 0
[2,2] => [2,2]
 => [2]
 => [1,1]
 => 1
[3,1] => [3,1]
 => [1]
 => [1]
 => 0
[4] => [4]
 => []
 => []
 => ? = 1
[1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,1,3] => [3,1,1]
 => [1,1]
 => [2]
 => 0
[1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,2] => [2,2,1]
 => [2,1]
 => [3]
 => 0
[1,3,1] => [3,1,1]
 => [1,1]
 => [2]
 => 0
[1,4] => [4,1]
 => [1]
 => [1]
 => 0
[2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[2,1,2] => [2,2,1]
 => [2,1]
 => [3]
 => 0
[2,2,1] => [2,2,1]
 => [2,1]
 => [3]
 => 0
[2,3] => [3,2]
 => [2]
 => [1,1]
 => 1
[3,1,1] => [3,1,1]
 => [1,1]
 => [2]
 => 0
[3,2] => [3,2]
 => [2]
 => [1,1]
 => 1
[4,1] => [4,1]
 => [1]
 => [1]
 => 0
[5] => [5]
 => []
 => []
 => ? = 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [3,2]
 => 2
[1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1
[1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,1,4] => [4,1,1]
 => [1,1]
 => [2]
 => 0
[1,2,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,2,1,2] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1
[1,2,2,1] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1
[1,2,3] => [3,2,1]
 => [2,1]
 => [3]
 => 0
[1,3,1,1] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,3,2] => [3,2,1]
 => [2,1]
 => [3]
 => 0
[1,4,1] => [4,1,1]
 => [1,1]
 => [2]
 => 0
[1,5] => [5,1]
 => [1]
 => [1]
 => 0
[2,1,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[2,1,1,2] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1
[2,1,2,1] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1
[2,1,3] => [3,2,1]
 => [2,1]
 => [3]
 => 0
[2,2,1,1] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 1
[2,2,2] => [2,2,2]
 => [2,2]
 => [4]
 => 0
[2,3,1] => [3,2,1]
 => [2,1]
 => [3]
 => 0
[2,4] => [4,2]
 => [2]
 => [1,1]
 => 1
[6] => [6]
 => []
 => []
 => ? = 0
Description
The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St001597
(load all 2 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
 => [[1],[]]
 => 1 = 0 + 1
[1,1] => [1,0,1,0]
 => [[1,1],[]]
 => 1 = 0 + 1
[2] => [1,1,0,0]
 => [[2],[]]
 => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
 => [[2,1],[]]
 => 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => 1 = 0 + 1
[3] => [1,1,1,0,0,0]
 => [[2,2],[]]
 => 2 = 1 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => 2 = 1 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => 2 = 1 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 1 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1,1],[]]
 => 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1,1],[]]
 => 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1,1],[]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1]]
 => 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2,1],[1]]
 => 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3,1],[2]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[2,2,2,2,2],[1,1,1,1]]
 => 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[3,2,2,2],[1,1,1]]
 => 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1,1]]
 => 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1,1]]
 => 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[3,3,3,2],[2,2,1]]
 => 1 = 0 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => 1 = 0 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1,1]]
 => 2 = 1 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[4,4,2],[1,1]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [[3,3,3,3],[2,2]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[4,3,3],[2]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4],[]]
 => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
Description
The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St000620
(load all 6 compositions to match this statistic)
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 78%●distinct values known / distinct values provided: 67%
Values
[1] => ([],1)
 => []
 => ? = 0
[1,1] => ([(0,1)],2)
 => [1]
 => ? ∊ {0,0}
[2] => ([],2)
 => []
 => ? ∊ {0,0}
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
[1,2] => ([(1,2)],3)
 => [1]
 => ? ∊ {0,0}
[2,1] => ([(0,2),(1,2)],3)
 => [2]
 => 0
[3] => ([],3)
 => []
 => ? ∊ {0,0}
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
[1,3] => ([(2,3)],4)
 => [1]
 => ? ∊ {0,0}
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => [2]
 => 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
[4] => ([],4)
 => []
 => ? ∊ {0,0}
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 0
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 0
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[1,4] => ([(3,4)],5)
 => [1]
 => ? ∊ {0,1}
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0
[2,3] => ([(2,4),(3,4)],5)
 => [2]
 => 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 0
[5] => ([],5)
 => []
 => ? ∊ {0,1}
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [15]
 => ? ∊ {1,1,1,1,2}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 0
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => 0
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => [3]
 => 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? ∊ {1,1,1,1,2}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 0
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => 0
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 0
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,5] => ([(4,5)],6)
 => [1]
 => ? ∊ {1,1,1,1,2}
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? ∊ {1,1,1,1,2}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 0
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[2,4] => ([(3,5),(4,5)],6)
 => [2]
 => 0
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => 0
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => [3]
 => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => 1
[6] => ([],6)
 => []
 => ? ∊ {1,1,1,1,2}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd. The case of an even minimum is [[St000621]].
Matching statistic: St000621
(load all 8 compositions to match this statistic)
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 78%●distinct values known / distinct values provided: 67%
Values
[1] => ([],1)
 => []
 => ? = 0
[1,1] => ([(0,1)],2)
 => [1]
 => ? ∊ {0,0}
[2] => ([],2)
 => []
 => ? ∊ {0,0}
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
[1,2] => ([(1,2)],3)
 => [1]
 => ? ∊ {0,0}
[2,1] => ([(0,2),(1,2)],3)
 => [2]
 => 1
[3] => ([],3)
 => []
 => ? ∊ {0,0}
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[1,3] => ([(2,3)],4)
 => [1]
 => ? ∊ {0,0}
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 0
[2,2] => ([(1,3),(2,3)],4)
 => [2]
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 0
[4] => ([],4)
 => []
 => ? ∊ {0,0}
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[1,4] => ([(3,4)],5)
 => [1]
 => ? ∊ {0,1}
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 0
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
[2,3] => ([(2,4),(3,4)],5)
 => [2]
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 1
[5] => ([],5)
 => []
 => ? ∊ {0,1}
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [15]
 => ? ∊ {0,1,1,1,2}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => 0
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 0
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => [3]
 => 0
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? ∊ {0,1,1,1,2}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 0
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 0
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
[1,5] => ([(4,5)],6)
 => [1]
 => ? ∊ {0,1,1,1,2}
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? ∊ {0,1,1,1,2}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 0
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 0
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => 0
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 0
[2,4] => ([(3,5),(4,5)],6)
 => [2]
 => 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 0
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => [3]
 => 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 0
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => 0
[6] => ([],6)
 => []
 => ? ∊ {0,1,1,1,2}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].
The following 178 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001592The maximal number of simple paths between any two different vertices of a graph. St000145The Dyson rank of a partition. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000534The number of 2-rises of a permutation. St000929The constant term of the character polynomial of an integer partition. St001587Half of the largest even part of an integer partition. St000640The rank of the largest boolean interval in a poset. St000137The Grundy value of an integer partition. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000340The number of non-final maximal constant sub-paths of length greater than one. St000389The number of runs of ones of odd length in a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000897The number of different multiplicities of parts of an integer partition. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001092The number of distinct even parts of a partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001383The BG-rank of an integer partition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000307The number of rowmotion orbits of a poset. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001323The independence gap of a graph. St000664The number of right ropes of a permutation. St000260The radius of a connected graph. St001871The number of triconnected components of a graph. St000259The diameter of a connected graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001728The number of invisible descents of a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001128The exponens consonantiae of a partition. St001845The number of join irreducibles minus the rank of a lattice. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000781The number of proper colouring schemes of a Ferrers diagram. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001525The number of symmetric hooks on the diagonal of a partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000031The number of cycles in the cycle decomposition of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. St001651The Frankl number of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000456The monochromatic index of a connected graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000761The number of ascents in an integer composition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001432The order dimension of the partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001961The sum of the greatest common divisors of all pairs of parts. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000091The descent variation of a composition. St000562The number of internal points of a set partition. St000709The number of occurrences of 14-2-3 or 14-3-2. St001130The number of two successive successions in a permutation. St001470The cyclic holeyness of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001857The number of edges in the reduced word graph of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001344The neighbouring number of a permutation. St001896The number of right descents of a signed permutations. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001964The interval resolution global dimension of a poset. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph.