Your data matches 196 different statistics following compositions of up to 3 maps.
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Matching statistic: St001588
(load all 3 compositions to match this statistic)
Mapping
St001588: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 1
[3,3]
 => 0
[3,2,1]
 => 1
[3,1,1,1]
 => 0
[2,2,2]
 => 0
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
[7]
 => 0
[6,1]
 => 1
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 1
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 0
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 1
[5,3]
 => 0
[5,2,1]
 => 1
Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Matching statistic: St001092
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => []
 => []
 => 0
[2]
 => []
 => []
 => 0
[1,1]
 => [1]
 => [1]
 => 0
[3]
 => []
 => []
 => 0
[2,1]
 => [1]
 => [1]
 => 0
[1,1,1]
 => [1,1]
 => [2]
 => 1
[4]
 => []
 => []
 => 0
[3,1]
 => [1]
 => [1]
 => 0
[2,2]
 => [2]
 => [1,1]
 => 0
[2,1,1]
 => [1,1]
 => [2]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[5]
 => []
 => []
 => 0
[4,1]
 => [1]
 => [1]
 => 0
[3,2]
 => [2]
 => [1,1]
 => 0
[3,1,1]
 => [1,1]
 => [2]
 => 1
[2,2,1]
 => [2,1]
 => [2,1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
[6]
 => []
 => []
 => 0
[5,1]
 => [1]
 => [1]
 => 0
[4,2]
 => [2]
 => [1,1]
 => 0
[4,1,1]
 => [1,1]
 => [2]
 => 1
[3,3]
 => [3]
 => [1,1,1]
 => 0
[3,2,1]
 => [2,1]
 => [2,1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[2,2,2]
 => [2,2]
 => [2,2]
 => 1
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 0
[7]
 => []
 => []
 => 0
[6,1]
 => [1]
 => [1]
 => 0
[5,2]
 => [2]
 => [1,1]
 => 0
[5,1,1]
 => [1,1]
 => [2]
 => 1
[4,3]
 => [3]
 => [1,1,1]
 => 0
[4,2,1]
 => [2,1]
 => [2,1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => 1
[3,2,2]
 => [2,2]
 => [2,2]
 => 1
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => 1
[8]
 => []
 => []
 => 0
[7,1]
 => [1]
 => [1]
 => 0
[6,2]
 => [2]
 => [1,1]
 => 0
[6,1,1]
 => [1,1]
 => [2]
 => 1
[5,3]
 => [3]
 => [1,1,1]
 => 0
[5,2,1]
 => [2,1]
 => [2,1]
 => 1
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St000291
(load all 2 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 => 1 => 0
[2]
 => 0 => 0 => 0 => 0
[1,1]
 => 11 => 11 => 11 => 0
[3]
 => 1 => 1 => 1 => 0
[2,1]
 => 01 => 01 => 10 => 1
[1,1,1]
 => 111 => 111 => 111 => 0
[4]
 => 0 => 0 => 0 => 0
[3,1]
 => 11 => 11 => 11 => 0
[2,2]
 => 00 => 00 => 00 => 0
[2,1,1]
 => 011 => 011 => 101 => 1
[1,1,1,1]
 => 1111 => 1111 => 1111 => 0
[5]
 => 1 => 1 => 1 => 0
[4,1]
 => 01 => 01 => 10 => 1
[3,2]
 => 10 => 10 => 01 => 0
[3,1,1]
 => 111 => 111 => 111 => 0
[2,2,1]
 => 001 => 001 => 100 => 1
[2,1,1,1]
 => 0111 => 0111 => 1011 => 1
[1,1,1,1,1]
 => 11111 => 11111 => 11111 => 0
[6]
 => 0 => 0 => 0 => 0
[5,1]
 => 11 => 11 => 11 => 0
[4,2]
 => 00 => 00 => 00 => 0
[4,1,1]
 => 011 => 011 => 101 => 1
[3,3]
 => 11 => 11 => 11 => 0
[3,2,1]
 => 101 => 101 => 110 => 1
[3,1,1,1]
 => 1111 => 1111 => 1111 => 0
[2,2,2]
 => 000 => 000 => 000 => 0
[2,2,1,1]
 => 0011 => 0011 => 1001 => 1
[2,1,1,1,1]
 => 01111 => 01111 => 10111 => 1
[1,1,1,1,1,1]
 => 111111 => 111111 => 111111 => 0
[7]
 => 1 => 1 => 1 => 0
[6,1]
 => 01 => 01 => 10 => 1
[5,2]
 => 10 => 10 => 01 => 0
[5,1,1]
 => 111 => 111 => 111 => 0
[4,3]
 => 01 => 01 => 10 => 1
[4,2,1]
 => 001 => 001 => 100 => 1
[4,1,1,1]
 => 0111 => 0111 => 1011 => 1
[3,3,1]
 => 111 => 111 => 111 => 0
[3,2,2]
 => 100 => 010 => 001 => 0
[3,2,1,1]
 => 1011 => 1011 => 1101 => 1
[3,1,1,1,1]
 => 11111 => 11111 => 11111 => 0
[2,2,2,1]
 => 0001 => 0001 => 1000 => 1
[2,2,1,1,1]
 => 00111 => 00111 => 10011 => 1
[2,1,1,1,1,1]
 => 011111 => 011111 => 101111 => 1
[1,1,1,1,1,1,1]
 => 1111111 => 1111111 => 1111111 => 0
[8]
 => 0 => 0 => 0 => 0
[7,1]
 => 11 => 11 => 11 => 0
[6,2]
 => 00 => 00 => 00 => 0
[6,1,1]
 => 011 => 011 => 101 => 1
[5,3]
 => 11 => 11 => 11 => 0
[5,2,1]
 => 101 => 101 => 110 => 1
Description
The number of descents of a binary word.
Matching statistic: St000875
(load all 2 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 => 1 => 0
[2]
 => 0 => 0 => 0 => 0
[1,1]
 => 11 => 11 => 11 => 0
[3]
 => 1 => 1 => 1 => 0
[2,1]
 => 01 => 01 => 10 => 1
[1,1,1]
 => 111 => 111 => 111 => 0
[4]
 => 0 => 0 => 0 => 0
[3,1]
 => 11 => 11 => 11 => 0
[2,2]
 => 00 => 00 => 00 => 0
[2,1,1]
 => 011 => 011 => 101 => 1
[1,1,1,1]
 => 1111 => 1111 => 1111 => 0
[5]
 => 1 => 1 => 1 => 0
[4,1]
 => 01 => 01 => 10 => 1
[3,2]
 => 10 => 10 => 01 => 0
[3,1,1]
 => 111 => 111 => 111 => 0
[2,2,1]
 => 001 => 001 => 100 => 1
[2,1,1,1]
 => 0111 => 0111 => 1011 => 1
[1,1,1,1,1]
 => 11111 => 11111 => 11111 => 0
[6]
 => 0 => 0 => 0 => 0
[5,1]
 => 11 => 11 => 11 => 0
[4,2]
 => 00 => 00 => 00 => 0
[4,1,1]
 => 011 => 011 => 101 => 1
[3,3]
 => 11 => 11 => 11 => 0
[3,2,1]
 => 101 => 101 => 110 => 1
[3,1,1,1]
 => 1111 => 1111 => 1111 => 0
[2,2,2]
 => 000 => 000 => 000 => 0
[2,2,1,1]
 => 0011 => 0011 => 1001 => 1
[2,1,1,1,1]
 => 01111 => 01111 => 10111 => 1
[1,1,1,1,1,1]
 => 111111 => 111111 => 111111 => 0
[7]
 => 1 => 1 => 1 => 0
[6,1]
 => 01 => 01 => 10 => 1
[5,2]
 => 10 => 10 => 01 => 0
[5,1,1]
 => 111 => 111 => 111 => 0
[4,3]
 => 01 => 01 => 10 => 1
[4,2,1]
 => 001 => 001 => 100 => 1
[4,1,1,1]
 => 0111 => 0111 => 1011 => 1
[3,3,1]
 => 111 => 111 => 111 => 0
[3,2,2]
 => 100 => 010 => 001 => 0
[3,2,1,1]
 => 1011 => 1011 => 1101 => 1
[3,1,1,1,1]
 => 11111 => 11111 => 11111 => 0
[2,2,2,1]
 => 0001 => 0001 => 1000 => 1
[2,2,1,1,1]
 => 00111 => 00111 => 10011 => 1
[2,1,1,1,1,1]
 => 011111 => 011111 => 101111 => 1
[1,1,1,1,1,1,1]
 => 1111111 => 1111111 => 1111111 => 0
[8]
 => 0 => 0 => 0 => 0
[7,1]
 => 11 => 11 => 11 => 0
[6,2]
 => 00 => 00 => 00 => 0
[6,1,1]
 => 011 => 011 => 101 => 1
[5,3]
 => 11 => 11 => 11 => 0
[5,2,1]
 => 101 => 101 => 110 => 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001115
(load all 3 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => []
 => []
 => [] => 0
[2]
 => []
 => []
 => [] => 0
[1,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[3]
 => []
 => []
 => [] => 0
[2,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[4]
 => []
 => []
 => [] => 0
[3,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[2,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[5]
 => []
 => []
 => [] => 0
[4,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[3,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[6]
 => []
 => []
 => [] => 0
[5,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
[7]
 => []
 => []
 => [] => 0
[6,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 1
[8]
 => []
 => []
 => [] => 0
[7,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
Description
The number of even descents of a permutation.
Matching statistic: St001421
(load all 2 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001421: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 => 1 => 0
[2]
 => 0 => 0 => 0 => 0
[1,1]
 => 11 => 11 => 11 => 0
[3]
 => 1 => 1 => 1 => 0
[2,1]
 => 01 => 01 => 10 => 1
[1,1,1]
 => 111 => 111 => 111 => 0
[4]
 => 0 => 0 => 0 => 0
[3,1]
 => 11 => 11 => 11 => 0
[2,2]
 => 00 => 00 => 00 => 0
[2,1,1]
 => 011 => 011 => 101 => 1
[1,1,1,1]
 => 1111 => 1111 => 1111 => 0
[5]
 => 1 => 1 => 1 => 0
[4,1]
 => 01 => 01 => 10 => 1
[3,2]
 => 10 => 10 => 01 => 0
[3,1,1]
 => 111 => 111 => 111 => 0
[2,2,1]
 => 001 => 001 => 100 => 1
[2,1,1,1]
 => 0111 => 0111 => 1011 => 1
[1,1,1,1,1]
 => 11111 => 11111 => 11111 => 0
[6]
 => 0 => 0 => 0 => 0
[5,1]
 => 11 => 11 => 11 => 0
[4,2]
 => 00 => 00 => 00 => 0
[4,1,1]
 => 011 => 011 => 101 => 1
[3,3]
 => 11 => 11 => 11 => 0
[3,2,1]
 => 101 => 101 => 110 => 1
[3,1,1,1]
 => 1111 => 1111 => 1111 => 0
[2,2,2]
 => 000 => 000 => 000 => 0
[2,2,1,1]
 => 0011 => 0011 => 1001 => 1
[2,1,1,1,1]
 => 01111 => 01111 => 10111 => 1
[1,1,1,1,1,1]
 => 111111 => 111111 => 111111 => 0
[7]
 => 1 => 1 => 1 => 0
[6,1]
 => 01 => 01 => 10 => 1
[5,2]
 => 10 => 10 => 01 => 0
[5,1,1]
 => 111 => 111 => 111 => 0
[4,3]
 => 01 => 01 => 10 => 1
[4,2,1]
 => 001 => 001 => 100 => 1
[4,1,1,1]
 => 0111 => 0111 => 1011 => 1
[3,3,1]
 => 111 => 111 => 111 => 0
[3,2,2]
 => 100 => 010 => 001 => 0
[3,2,1,1]
 => 1011 => 1011 => 1101 => 1
[3,1,1,1,1]
 => 11111 => 11111 => 11111 => 0
[2,2,2,1]
 => 0001 => 0001 => 1000 => 1
[2,2,1,1,1]
 => 00111 => 00111 => 10011 => 1
[2,1,1,1,1,1]
 => 011111 => 011111 => 101111 => 1
[1,1,1,1,1,1,1]
 => 1111111 => 1111111 => 1111111 => 0
[8]
 => 0 => 0 => 0 => 0
[7,1]
 => 11 => 11 => 11 => 0
[6,2]
 => 00 => 00 => 00 => 0
[6,1,1]
 => 011 => 011 => 101 => 1
[5,3]
 => 11 => 11 => 11 => 0
[5,2,1]
 => 101 => 101 => 110 => 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St000256
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1]
 => []
 => []
 => ?
 => ? = 0
[2]
 => []
 => []
 => ?
 => ? = 0
[1,1]
 => [1]
 => [1]
 => [1]
 => 0
[3]
 => []
 => []
 => ?
 => ? = 0
[2,1]
 => [1]
 => [1]
 => [1]
 => 0
[1,1,1]
 => [1,1]
 => [2]
 => [2]
 => 1
[4]
 => []
 => []
 => ?
 => ? = 0
[3,1]
 => [1]
 => [1]
 => [1]
 => 0
[2,2]
 => [2]
 => [1,1]
 => [1,1]
 => 0
[2,1,1]
 => [1,1]
 => [2]
 => [2]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [3]
 => [2,1]
 => 0
[5]
 => []
 => []
 => ?
 => ? = 0
[4,1]
 => [1]
 => [1]
 => [1]
 => 0
[3,2]
 => [2]
 => [1,1]
 => [1,1]
 => 0
[3,1,1]
 => [1,1]
 => [2]
 => [2]
 => 1
[2,2,1]
 => [2,1]
 => [2,1]
 => [3]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => [2,1]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[6]
 => []
 => []
 => ?
 => ? = 0
[5,1]
 => [1]
 => [1]
 => [1]
 => 0
[4,2]
 => [2]
 => [1,1]
 => [1,1]
 => 0
[4,1,1]
 => [1,1]
 => [2]
 => [2]
 => 1
[3,3]
 => [3]
 => [1,1,1]
 => [1,1,1]
 => 0
[3,2,1]
 => [2,1]
 => [2,1]
 => [3]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => [2,1]
 => 0
[2,2,2]
 => [2,2]
 => [2,2]
 => [4]
 => 1
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [2,2,1]
 => 0
[7]
 => []
 => []
 => ?
 => ? = 0
[6,1]
 => [1]
 => [1]
 => [1]
 => 0
[5,2]
 => [2]
 => [1,1]
 => [1,1]
 => 0
[5,1,1]
 => [1,1]
 => [2]
 => [2]
 => 1
[4,3]
 => [3]
 => [1,1,1]
 => [1,1,1]
 => 0
[4,2,1]
 => [2,1]
 => [2,1]
 => [3]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => [2,1]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [3,1]
 => 1
[3,2,2]
 => [2,2]
 => [2,2]
 => [4]
 => 1
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [4,1]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [3,2]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [2,2,1]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => [2,2,2]
 => 1
[8]
 => []
 => []
 => ?
 => ? = 0
[7,1]
 => [1]
 => [1]
 => [1]
 => 0
[6,2]
 => [2]
 => [1,1]
 => [1,1]
 => 0
[6,1,1]
 => [1,1]
 => [2]
 => [2]
 => 1
[5,3]
 => [3]
 => [1,1,1]
 => [1,1,1]
 => 0
[5,2,1]
 => [2,1]
 => [2,1]
 => [3]
 => 1
[5,1,1,1]
 => [1,1,1]
 => [3]
 => [2,1]
 => 0
[4,4]
 => [4]
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [3,1]
 => 1
[4,2,2]
 => [2,2]
 => [2,2]
 => [4]
 => 1
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[3,3,2]
 => [3,2]
 => [2,2,1]
 => [5]
 => 1
[3,3,1,1]
 => [3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => 0
[9]
 => []
 => []
 => ?
 => ? = 0
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000257
(load all 3 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1]
 => []
 => []
 => ?
 => ? = 0
[2]
 => []
 => []
 => ?
 => ? = 0
[1,1]
 => [1]
 => [1]
 => [1]
 => 0
[3]
 => []
 => []
 => ?
 => ? = 0
[2,1]
 => [1]
 => [1]
 => [1]
 => 0
[1,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[4]
 => []
 => []
 => ?
 => ? = 0
[3,1]
 => [1]
 => [1]
 => [1]
 => 0
[2,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[2,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[5]
 => []
 => []
 => ?
 => ? = 0
[4,1]
 => [1]
 => [1]
 => [1]
 => 0
[3,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[3,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[2,2,1]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[6]
 => []
 => []
 => ?
 => ? = 0
[5,1]
 => [1]
 => [1]
 => [1]
 => 0
[4,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[4,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[3,3]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[3,2,1]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[2,2,2]
 => [2,2]
 => [2,2]
 => [1,1,1,1]
 => 1
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [5]
 => 0
[7]
 => []
 => []
 => ?
 => ? = 0
[6,1]
 => [1]
 => [1]
 => [1]
 => 0
[5,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[5,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[4,3]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[4,2,1]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[3,2,2]
 => [2,2]
 => [2,2]
 => [1,1,1,1]
 => 1
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [3,1,1]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [2,2,1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [5]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => [3,3]
 => 1
[8]
 => []
 => []
 => ?
 => ? = 0
[7,1]
 => [1]
 => [1]
 => [1]
 => 0
[6,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[6,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[5,3]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[5,2,1]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[5,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[4,4]
 => [4]
 => [1,1,1,1]
 => [4]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[4,2,2]
 => [2,2]
 => [2,2]
 => [1,1,1,1]
 => 1
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[3,3,2]
 => [3,2]
 => [2,2,1]
 => [1,1,1,1,1]
 => 1
[3,3,1,1]
 => [3,1,1]
 => [3,1,1]
 => [3,2]
 => 0
[9]
 => []
 => []
 => ?
 => ? = 0
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St001732
(load all 4 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1 = 0 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 0 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 0 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,2,2,1]
 => [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]
 => 2 = 1 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,2} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,2} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 1 = 0 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 1 = 0 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,2} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,1,1,1,1} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,1,1,1,1} + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,1,1} + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,1} + 1
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,1,1,1,1} + 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,1,1,1,1} + 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: St000779
(load all 4 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => [1,3,2] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => [1,2,3] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [1,4,2,3] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [1,3,2,4] => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [1,3,4,5,2] => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [1,2,5,3,4] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,3,4,2] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [1,4,2,3] => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [1,4,2,3] => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [1,3,4,2,5] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [1,3,4,5,6,2] => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 0
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,6,4] => [1,2,3,6,4,5] => 0
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [1,4,2,5,3] => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [1,2,5,3,4] => 0
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,2,3,4] => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [1,5,2,3,4] => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [1,3,5,2,4] => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => [1,3,4,5,2,6] => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => ? ∊ {0,0}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,7,5] => [1,2,3,4,7,5,6] => 0
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,2,3,6,4] => [1,5,2,3,6,4] => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [4,1,2,3,6,5] => [1,2,3,6,4,5] => 0
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,2,4,5,3] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [1,5,2,3,4] => 0
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [1,3,5,2,4] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [1,4,2,5,3] => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,3,5,2,4] => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [1,3,2,4,5] => 0
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,1,6,3,4,5] => [1,6,2,3,4,5] => 0
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [1,4,5,2,3] => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => [1,3,4,6,2,5] => 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,1,3,4,5,6] => [1,3,4,5,6,2,7] => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => [1,3,4,5,6,7,8,2] => ? ∊ {0,0}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => ? ∊ {0,0,0,0,1,1}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,8,6] => [1,2,3,4,5,8,6,7] => ? ∊ {0,0,0,0,1,1}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => [1,6,2,3,4,7,5] => ? ∊ {0,0,0,0,1,1}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,7,6] => [1,2,3,4,7,5,6] => 0
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4] => [1,2,5,3,6,4] => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,5,1,2,6,3] => [1,2,6,3,4,5] => 0
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,6,5] => [1,2,4,6,3,5] => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [1,4,5,2,3] => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [1,5,2,4,3] => 0
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,1,7,3,4,5,6] => [1,7,2,3,4,5,6] => ? ∊ {0,0,0,0,1,1}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7] => [1,3,4,5,6,7,2,8] => ? ∊ {0,0,0,0,1,1}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2] => ? ∊ {0,0,0,0,1,1}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => ? ∊ {0,0,0,0,1,1,1,1}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => [1,2,3,4,5,6,9,7,8] => ? ∊ {0,0,0,0,1,1,1,1}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,7,2,3,4,5,8,6] => [1,7,2,3,4,5,8,6] => ? ∊ {0,0,0,0,1,1,1,1}
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,8,7] => [1,2,3,4,5,8,6,7] => ? ∊ {0,0,0,0,1,1,1,1}
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2,1,8,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => ? ∊ {0,0,0,0,1,1,1,1}
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6,8] => [1,3,4,5,6,8,2,7] => ? ∊ {0,0,0,0,1,1,1,1}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [2,9,1,3,4,5,6,7,8] => [1,3,4,5,6,7,8,2,9] => ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9,10] => [1,3,4,5,6,7,8,9,10,2] => ? ∊ {0,0,0,0,1,1,1,1}
Description
The tier of a permutation. This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$. According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
The following 186 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000754The Grundy value for the game of removing nestings in a perfect matching. St000664The number of right ropes of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001728The number of invisible descents of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000886The number of permutations with the same antidiagonal sums. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000872The number of very big descents of a permutation. St001487The number of inner corners of a skew partition. St000259The diameter of a connected graph. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000252The number of nodes of degree 3 of a binary tree. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001549The number of restricted non-inversions between exceedances. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000137The Grundy value of an integer partition. St000143The largest repeated part of a partition. St000389The number of runs of ones of odd length in a binary word. St000481The number of upper covers of a partition in dominance order. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St001280The number of parts of an integer partition that are at least two. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000663The number of right floats of a permutation. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard 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. St001939The number of parts that are equal to their multiplicity in the integer partition. St000352The Elizalde-Pak rank of a permutation. St000007The number of saliances of the permutation. St000842The breadth of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000850The number of 1/2-balanced pairs in a poset. St000938The number of zeros of the symmetric group character corresponding to the partition. St001399The distinguishing number of a poset. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000649The number of 3-excedences of a permutation. 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. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001498The normalised height of a Nakayama algebra with magnitude 1. St000617The number of global maxima of a Dyck path. St001162The minimum jump of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001513The number of nested exceedences of a permutation. St001665The number of pure excedances of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000650The number of 3-rises of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000260The radius of a connected graph. St000353The number of inner valleys 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. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001202Call 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. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001729The number of visible descents of a permutation. St000092The number of outer peaks of a permutation. St000542The number of left-to-right-minima of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. 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 $eA$. St001095The number of non-isomorphic posets with precisely one further covering relation. St000633The size of the automorphism group of a poset. St001651The Frankl number of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000058The order of a permutation. St000317The cycle descent number of a permutation. St000552The number of cut vertices of a graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001128The exponens consonantiae of a partition. St001964The interval resolution global dimension of a poset. St000741The Colin de Verdière graph invariant. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001820The size of the image of the pop stack sorting operator. St000455The second largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000768The number of peaks in an integer composition. St001730The number of times the path corresponding to a binary word crosses the base line. St001811The Castelnuovo-Mumford regularity of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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 $g$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000527The width of the poset. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001423The number of distinct cubes in a binary word. St000902 The minimal number of repetitions of an integer composition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001884The number of borders of a binary word. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000758The length of the longest staircase fitting into an integer composition. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St000068The number of minimal elements in a poset. St000534The number of 2-rises of a permutation. St000451The length of the longest pattern of the form k 1 2. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000679The pruning number of an ordered tree.