searching the database
Your data matches 100 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000007
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2 = 3 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 2 = 3 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 2 - 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000745
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [[1],[2]]
=> 2 = 3 - 1
[2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
[1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 2 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 3 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 1 = 2 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 2 = 3 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 2 = 3 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 1 = 2 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 1 = 2 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 2 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001159
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001159: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001159: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 3 - 2
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 2 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 3 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 3 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 2 - 2
Description
Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St001217
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 1 = 3 - 2
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 3 - 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 3 - 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0 = 2 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 3 - 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> 10 => 00 => 3
[2]
=> [1,0,1,0]
=> 1010 => 0010 => 3
[1,1]
=> [1,1,0,0]
=> 1100 => 0100 => 2
[3]
=> [1,0,1,0,1,0]
=> 101010 => 001010 => 3
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => 001100 => 3
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 010100 => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 00101010 => 3
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 00101100 => 3
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => 011000 => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00110100 => 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 01010100 => 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0010101010 => 3
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0010101100 => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00111000 => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 01100100 => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 01101000 => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 01110000 => 2
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000439
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000025
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000069
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => ([],2)
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 3 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1 = 2 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2 = 3 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
Description
The number of maximal elements of a poset.
Matching statistic: St000314
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,2] => 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => 2 = 3 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => 2 = 3 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => 1 = 2 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [5,6,4,3,2,1] => 2 = 3 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,5,2,1,4] => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 1 = 2 - 1
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000542
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 3 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2 = 3 - 1
[1,1,1,1]
=> [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] => 1 = 2 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 2 = 3 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[2,2,2]
=> [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] => 1 = 2 - 1
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
The following 90 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000733The row containing the largest entry of a standard tableau. St000991The number of right-to-left minima of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000237The number of small exceedances. St000297The number of leading ones in a binary word. St000352The Elizalde-Pak rank of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000989The number of final rises of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001271The competition number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000061The number of nodes on the left branch of a binary tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000504The cardinality of the first block of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000990The first ascent of a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000390The number of runs of ones in a binary word. St000461The rix statistic of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001593This is the number of standard Young tableaux of the given shifted shape. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000260The radius of a connected graph. St000658The number of rises of length 2 of a Dyck path. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000239The number of small weak excedances. St000929The constant term of the character polynomial 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. St001525The number of symmetric hooks on the diagonal of a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000045The number of linear extensions of a binary tree. St000782The indicator function of whether a given perfect matching is an L & P matching. St001050The number of terminal closers of a set partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001722The number of minimal chains with small intervals between a binary word and the top element. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000090The variation of a composition. St000284The Plancherel distribution on integer partitions. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. 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. St001128The exponens consonantiae of a partition. St001399The distinguishing number of a poset. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001651The Frankl number of a lattice. St000941The number of characters of the symmetric group whose value on the partition is even. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!