- FindStat

Your data matches 60 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000662

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => [] => 0
[1,0,1,0]
 => [2,1] => [2,1] => [1] => 0
[1,1,0,0]
 => [1,2] => [1,2] => [1] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [2,1] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,2] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [3,2,1] => 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,1,2] => [3,1,2] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,1,3,2] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,4,2,3] => [1,2,3] => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1,4,2] => [3,1,2] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => [3,1,2] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,4,3,1,2] => [4,3,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,4,1,3,2] => [4,1,3,2] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,2,1,3] => [4,2,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,1,4,3,2] => [1,4,3,2] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,1,4,2,3] => [1,4,2,3] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,1,4,2] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,1,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,5,4,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,5,2,4,3] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,5,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,5,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1,5,3,2] => [4,1,3,2] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,1,5,2,3] => [4,1,2,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,1,5,2] => [4,3,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,3,1,2,5] => [4,3,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [4,1,2,5,3] => [4,1,2,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1

Description

The staircase size of the code of a permutation. The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ . The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ . This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

Matching statistic: St000288

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 98%●distinct values known / distinct values provided: 90%

Values

[1,0]
 => [1] =>  => ? => ? = 0 + 1
[1,0,1,0]
 => [2,1] => 1 => 1 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => 0 => 1 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => 11 => 11 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [2,3,1] => 01 => 10 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => 01 => 10 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => 10 => 11 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => 00 => 01 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 111 => 111 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 101 => 110 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 011 => 101 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 011 => 101 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 001 => 010 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 011 => 101 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 001 => 010 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 101 => 110 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 110 => 111 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 010 => 101 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 001 => 010 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 010 => 101 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 100 => 101 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 001 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1111 => 1111 => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 1101 => 1110 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 1011 => 1101 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 1011 => 1101 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 1001 => 1010 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 0111 => 1011 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 0101 => 1010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => 0111 => 1011 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 0111 => 1011 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 0011 => 0101 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 0011 => 0101 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 0011 => 0101 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 0101 => 1010 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0001 => 0010 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 0111 => 1011 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 0101 => 1010 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 0011 => 0101 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 0011 => 0101 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 0001 => 0010 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 1011 => 1101 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 1001 => 1010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 1101 => 1110 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 1110 => 1111 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 1010 => 1101 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 0101 => 1010 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => 0110 => 1011 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 0110 => 1011 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 0010 => 0101 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 0011 => 0101 => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => [5,6,1,2,3,4,7,8] => ? => ? => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [10,9,8,7,6,5,4,3,2,1,11] => 1111111110 => 1111111111 => ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [7,8,9,10,6,5,4,3,2,1] => 111110001 => 111110010 => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [6,7,8,9,10,5,4,3,2,1] => 111100001 => 111100010 => ? = 4 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [5,6,7,8,9,10,4,3,2,1] => 111000001 => 111000010 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [4,5,6,7,8,9,10,3,2,1] => 110000001 => 110000010 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [10,9,8,7,6,5,4,3,1,2] => 011111111 => 101111111 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [11,10,9,8,7,6,5,4,3,1,2] => 0111111111 => 1011111111 => ? = 8 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9,10] => 100000000 => 100000001 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9,10,11] => 1000000000 => 1000000001 => ? = 1 + 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [7,6,5,4,3,2,1,8,9,10] => 111111000 => 111111001 => ? = 6 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [6,5,4,3,2,1,7,8,9,10] => 111110000 => 111110001 => ? = 5 + 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,1,4,5,6,7,8,9,10] => 110000000 => 110000001 => ? = 2 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [5,4,3,2,1,6,7,8,9,10] => 111100000 => 111100001 => ? = 4 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [4,3,2,1,5,6,7,8,9,10] => 111000000 => 111000001 => ? = 3 + 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,4,5,6,7,8,9,1,10] => 0000000101 => 0000001010 => ? = 1 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000318
(load all 2 compositions to match this statistic)

Mapping

Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 93%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [1,0]
 => []
 => ?
 => ? = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [1]
 => []
 => 1 = 0 + 1
[1,1,0,0]
 => [1,1,0,0]
 => []
 => ?
 => ? = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => []
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => []
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => []
 => ?
 => ? = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => []
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => []
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => []
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => ? = 6 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,5,4,3,2,1]
 => [5,5,4,3,2,1]
 => ? = 5 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5,1]
 => ?
 => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
 => [6,5,5,4,1]
 => ?
 => ? = 3 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
 => [6,5,5,1]
 => [5,5,1]
 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2]
 => [6,5,4,3,2]
 => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,3,2,1]
 => [5,5,4,3,2,1]
 => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,3,2,1]
 => [4,4,4,3,2,1]
 => ? = 4 + 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,3,3,3,3,2,1]
 => [3,3,3,3,2,1]
 => ? = 3 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,3]
 => ?
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,3]
 => ?
 => ? = 3 + 1
[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,1,1]
 => ?
 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4]
 => [6,5,4]
 => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3]
 => ?
 => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3]
 => ?
 => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3]
 => ?
 => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,3]
 => ?
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,3,3,3]
 => [3,3,3,3]
 => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4,4,4,2]
 => ?
 => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,2]
 => [4,4,4,2]
 => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5]
 => [6,5]
 => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
 => [6,4,4,4]
 => ?
 => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4]
 => ?
 => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4]
 => ?
 => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4]
 => [4,4,4]
 => ? = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3]
 => ?
 => ? = 2 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => []
 => ?
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => [8,6,6]
 => ?
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,7,6,5,4,3,2,1]
 => [7,6,5,4,3,2,1]
 => ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,6,6,5,4,3,2,1]
 => [6,6,5,4,3,2,1]
 => ? = 6 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => ? = 6 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => []
 => ?
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1]
 => [7,6,5,4,3,2,1]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3,2,1]
 => [8,7,6,5,4,3,2,1]
 => ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,3,2,1]
 => [7,6,5,4,3,2,1]
 => ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,8,7,6,5,4,3,2,1]
 => [8,7,6,5,4,3,2,1]
 => ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [9,9,8,7,6,5,4,3,2,1]
 => [9,8,7,6,5,4,3,2,1]
 => ? = 9 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2]
 => [7,6,5,4,3,2]
 => ? = 6 + 1

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000996
(load all 2 compositions to match this statistic)

Mapping

Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%

Values

[1,0]
 => [1,0]
 => [1] => [] => 0
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => [1] => 0
[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => [1] => 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1] => 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => [1,2] => 0
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1] => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,3,1] => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [3,1,2] => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [3,1,2] => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,2,3] => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,3,1] => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2] => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [3,4,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,4,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,4,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [3,1,4,2] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,4,1,3] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,3,1,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [4,1,2,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [4,5,6,7,8,1,2,3] => [4,5,6,7,1,2,3] => ? = 4
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,7,8,1,2,4,5,6] => ? => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
 => [4,5,7,8,1,2,3,6] => [4,5,7,1,2,3,6] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
 => [6,1,2,7,8,3,4,5] => ? => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [6,1,2,7,3,4,8,5] => ? => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => [7,1,2,3,8,4,5,6] => ? => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
 => [4,1,2,5,6,8,3,7] => ? => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
 => [5,1,2,3,6,8,4,7] => ? => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => [5,1,6,8,2,3,4,7] => ? => ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [5,8,1,2,3,4,6,7] => ? => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [6,8,1,2,3,4,5,7] => ? => ? = 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [1,5,2,6,3,8,4,7] => [1,5,2,6,3,4,7] => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
 => [1,5,2,8,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
 => [1,5,6,2,8,3,4,7] => ? => ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
 => [1,6,2,3,4,8,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
 => [1,6,2,8,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
 => [1,6,8,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,3,1,8,4,5,6,7] => ? => ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,1,4,5,6,8,7] => ? => ? = 2
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,1,3,4,5,8,6,7] => ? => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [2,6,1,3,4,5,8,7] => ? => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [1,2,6,7,8,3,4,5] => ? => ? = 2
[1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,6,7,3,8,5] => ? => ? = 3
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [1,2,7,8,3,4,5,6] => ? => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => [1,2,7,3,8,4,5,6] => ? => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,6,3,7,8,5] => ? => ? = 3
[1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
 => [1,2,5,6,8,3,4,7] => ? => ? = 2
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
 => [1,2,6,8,3,4,5,7] => ? => ? = 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,3,2,4,5,6,8,7] => ? => ? = 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [3,1,2,8,4,5,6,7] => ? => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,7,1,2,8] => ? => ? = 5
[1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [3,1,2,4,7,8,5,6] => ? => ? = 2
[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,5,4,6,7,8] => ? => ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [3,1,2,4,5,6,8,7] => ? => ? = 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,5,3,7,8,6] => ? => ? = 3
[1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,5,6,3,8,7] => ? => ? = 3
[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,4,5,3,6,7,8] => ? => ? = 2
[1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,4,3,5,6,8,7] => ? => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => [4,1,2,3,7,8,5,6] => [4,1,2,3,7,5,6] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
 => [4,1,2,3,7,5,8,6] => [4,1,2,3,7,5,6] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
 => [4,1,5,2,3,8,6,7] => ? => ? = 2
[1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
 => [4,1,5,2,3,6,7,8] => ? => ? = 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3,8] => ? => ? = 4
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,2,3,5,4,6,8,7] => ? => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [1,2,5,3,4,7,8,6] => ? => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4,8,6,7] => ? => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
 => [5,1,2,3,4,7,8,6] => ? => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
 => [5,1,2,3,4,8,6,7] => ? => ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6,8] => ? => ? = 2

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

Matching statistic: St000159
(load all 2 compositions to match this statistic)

Mapping

Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => [1,0]
 => []
 => ?
 => ? = 0
[1,0,1,0]
 => [1,0,1,0]
 => [1]
 => []
 => 0
[1,1,0,0]
 => [1,1,0,0]
 => []
 => ?
 => ? = 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => []
 => 0
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => []
 => 0
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => []
 => ?
 => ? = 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => []
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => []
 => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => []
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? = 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1]
 => [5,4,3,2,1]
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2,1]
 => [4,3,3,2,1]
 => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1]
 => [5,4,3,1]
 => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,1]
 => [5,4,2,1,1]
 => ? = 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => [5,3,2,2,1]
 => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => [4,3,3,2,1]
 => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => [4,4,3,2,1]
 => ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2]
 => [5,4,3,2]
 => ? = 4
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2]
 => [4,4,3,2]
 => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,1]
 => [5,4,3,1,1]
 => ? = 4
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => [4,4,3,1,1]
 => ? = 3
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => [5,4,2,2,1]
 => ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => [4,4,2,2,1]
 => ? = 3
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => [5,3,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => [4,4,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => [5,4,3,2,1]
 => ? = 5
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [4,4,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => [5,3,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [4,3,3,2,1]
 => ? = 4
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => [4,4,2,2,1]
 => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => [5,4,2,2,1]
 => ? = 4
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => [4,4,2,2,1]
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => [5,3,2,2,1]
 => ? = 4
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1,1]
 => [5,3,3,1,1]
 => ? = 3
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1,1]
 => [4,4,3,1,1]
 => ? = 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1,1]
 => [5,4,3,1,1]
 => ? = 4
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,1,1]
 => [4,4,3,1,1]
 => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1,1]
 => [5,3,3,1,1]
 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,1,1]
 => [5,4,2,1,1]
 => ? = 4
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2]
 => [5,4,2,2]
 => ? = 3
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2]
 => [5,3,3,2]
 => ? = 3
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2]
 => [4,4,3,2]
 => ? = 3
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2]
 => [5,4,3,2]
 => ? = 4
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2]
 => [4,4,3,2]
 => ? = 3
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2]
 => [5,3,3,2]
 => ? = 3
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2]
 => [5,4,2,2]
 => ? = 3
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1]
 => [5,4,3,1]
 => ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => ?
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,1]
 => [5,4,3,2,1]
 => ? = 5
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,5,4,3,2,1]
 => [5,5,4,3,2,1]
 => ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5,1]
 => ?
 => ? = 3
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
 => [6,5,5,4,1]
 => ?
 => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2]
 => [6,5,4,3,2]
 => ? = 5

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Matching statistic: St001036
(load all 88 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [1,0]
 => []
 => []
 => ? = 0
[1,0,1,0]
 => [1,1,0,0]
 => []
 => []
 => ? = 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]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => []
 => ? = 0
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? = 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? = 0
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? = 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,2,2]
 => ?
 => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,1,1]
 => ?
 => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [6,6,3,1,1,1]
 => ?
 => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [7,5,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3,1]
 => ?
 => ? = 5
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [4,4,3,3,3,1]
 => ?
 => ? = 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,2,1]
 => ?
 => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,3,3,1]
 => ?
 => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [6,6,4,3,2,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,2]
 => ?
 => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4]
 => ?
 => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [7,6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 1
[1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3,1,1]
 => ?
 => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,5,2,2,1,1]
 => ?
 => ? = 5
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [7,4,2,2,2,1,1]
 => ?
 => ? = 4
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,3]
 => ?
 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,1,1]
 => ?
 => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1,1]
 => ?
 => ? = 3
[1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,2,2,1]
 => ?
 => ? = 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => []
 => []
 => ? = 0
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,1,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 3
[1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
 => [7,5,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 3
[1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
 => [7,5,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
 => ? = 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3,2,1]
 => ?
 => ? = 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [6,5,3,2,1,1,1]
 => ?
 => ? = 2
[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]
 => [7,6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [4,4,3,2,2,2,1]
 => ?
 => ? = 2
[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]
 => [8,6,5,4,3,2,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 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]
 => [8,7,6,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [8,7,6,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0

Description

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St001489

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 84%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [1] => [1] => [] => ? = 0
[1,0,1,0]
 => [2,1] => [2,1] => [1] => 0
[1,1,0,0]
 => [1,2] => [1,2] => [1] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [2,1] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,2] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [3,2,1] => 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,1,2] => [3,1,2] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,1,3,2] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,4,2,3] => [1,2,3] => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1,4,2] => [3,1,2] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => [3,1,2] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,4,3,1,2] => [4,3,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,4,1,3,2] => [4,1,3,2] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,2,1,3] => [4,2,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,1,4,3,2] => [1,4,3,2] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,1,4,2,3] => [1,4,2,3] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,1,4,2] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,1,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,5,4,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,5,2,4,3] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,5,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,5,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1,5,3,2] => [4,1,3,2] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,1,5,2,3] => [4,1,2,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,1,5,2] => [4,3,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,3,1,2,5] => [4,3,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [4,1,2,5,3] => [4,1,2,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ? = 2
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,8,2,1] => [8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => ? = 2
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [4,5,6,3,7,8,2,1] => [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,7,8,2,1] => [8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => ? = 3
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,8,2,1] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,2,3,1] => [8,1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 5
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,6,7,8,1] => [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 5
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,2,1,3] => [7,1,8,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,2,1,4] => [7,6,1,8,2,3,4,5] => [7,6,1,2,3,4,5] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,1,8] => [7,6,5,4,3,1,2,8] => [7,6,5,4,3,1,2] => ? = 5
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,3,2,1,8] => [7,6,5,4,1,2,3,8] => [7,6,5,4,1,2,3] => ? = 4
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,2,1,8] => [7,6,5,1,2,3,4,8] => [7,6,5,1,2,3,4] => ? = 3
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,2,1,7] => [7,6,1,2,3,4,8,5] => [7,6,1,2,3,4,5] => ? = 2
[1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,3,1,4] => [7,1,2,8,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,2,4,1,5] => [7,5,1,2,8,3,4,6] => [7,5,1,2,3,4,6] => ? = 2
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,1,8] => [7,5,1,2,3,4,6,8] => [7,5,1,2,3,4,6] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [7,8,2,3,4,5,1,6] => [7,1,2,3,4,8,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,6,1,7] => [7,3,1,2,4,5,8,6] => [7,3,1,2,4,5,6] => ? = 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,1,8] => [7,3,1,2,4,5,6,8] => [7,3,1,2,4,5,6] => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,6,1,7] => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,5,6] => ? = 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,1,8] => [7,2,1,3,4,5,6,8] => [7,2,1,3,4,5,6] => ? = 2
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1,8] => [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,1,2,4] => [1,7,2,8,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,1,2,5] => [1,7,2,3,8,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
 => [7,8,3,4,5,1,2,6] => [1,7,2,3,4,8,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,1,2,7] => [1,7,2,3,4,5,8,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,1,2,8] => [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,1,3,4] => [6,1,2,8,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7] => ? = 5
[1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
 => [7,8,6,2,3,1,4,5] => [6,1,2,3,8,7,4,5] => [6,1,2,3,7,4,5] => ? = 2
[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [5,6,7,2,3,1,4,8] => [6,1,2,7,3,4,5,8] => [6,1,2,7,3,4,5] => ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => [6,1,2,3,4,5,8,7] => [6,1,2,3,4,5,7] => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7,8] => [6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,1,2,4,6] => [1,6,2,7,3,8,4,5] => [1,6,2,7,3,4,5] => ? = 2
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
 => [5,6,7,3,1,2,4,8] => [1,6,2,7,3,4,5,8] => [1,6,2,7,3,4,5] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
 => [7,8,6,2,1,3,4,5] => [5,1,2,3,8,7,4,6] => [5,1,2,3,7,4,6] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [8,6,7,2,1,3,4,5] => [5,1,2,3,8,4,7,6] => [5,1,2,3,4,7,6] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
 => [6,7,8,2,1,3,4,5] => [5,1,2,3,8,4,6,7] => [5,1,2,3,4,6,7] => ? = 1

Description

The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].

Matching statistic: St000470

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 84%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [1] => [1] => [] => ? = 0 + 1
[1,0,1,0]
 => [2,1] => [2,1] => [1] => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [2,1] => 2 = 1 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,2] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [3,2,1] => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,1,2] => [3,1,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,1,3,2] => [1,3,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [2,1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,4,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1,4,2] => [3,1,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,3] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,4,3,1,2] => [4,3,1,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,4,1,3,2] => [4,1,3,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,1,2,3] => [4,1,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,1,4,2] => [3,1,4,2] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,1,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,5,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,5,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,5,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,5,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1,5,3,2] => [4,1,3,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,1,5,2,3] => [4,1,2,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,1,5,2] => [4,3,1,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,3,1,2,5] => [4,3,1,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [4,1,2,5,3] => [4,1,2,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => ? = 4 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,8,2,1] => [8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [4,5,6,3,7,8,2,1] => [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,7,8,2,1] => [8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => ? = 3 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,8,2,1] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,2,3,1] => [8,1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 5 + 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,6,7,8,1] => [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 3 + 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 2 + 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 5 + 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,2,1,3] => [7,1,8,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,2,1,4] => [7,6,1,8,2,3,4,5] => [7,6,1,2,3,4,5] => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,1,8] => [7,6,5,4,3,1,2,8] => [7,6,5,4,3,1,2] => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,3,2,1,8] => [7,6,5,4,1,2,3,8] => [7,6,5,4,1,2,3] => ? = 4 + 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,2,1,8] => [7,6,5,1,2,3,4,8] => [7,6,5,1,2,3,4] => ? = 3 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,2,1,7] => [7,6,1,2,3,4,8,5] => [7,6,1,2,3,4,5] => ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,3,1,4] => [7,1,2,8,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,2,4,1,5] => [7,5,1,2,8,3,4,6] => [7,5,1,2,3,4,6] => ? = 2 + 1
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,1,8] => [7,5,1,2,3,4,6,8] => [7,5,1,2,3,4,6] => ? = 2 + 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [7,8,2,3,4,5,1,6] => [7,1,2,3,4,8,5,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,6,1,7] => [7,3,1,2,4,5,8,6] => [7,3,1,2,4,5,6] => ? = 2 + 1
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,1,8] => [7,3,1,2,4,5,6,8] => [7,3,1,2,4,5,6] => ? = 2 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,6,1,7] => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,1,8] => [7,2,1,3,4,5,6,8] => [7,2,1,3,4,5,6] => ? = 2 + 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1,8] => [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,1,2,4] => [1,7,2,8,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,1,2,5] => [1,7,2,3,8,4,5,6] => [1,7,2,3,4,5,6] => ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
 => [7,8,3,4,5,1,2,6] => [1,7,2,3,4,8,5,6] => [1,7,2,3,4,5,6] => ? = 1 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,1,2,7] => [1,7,2,3,4,5,8,6] => [1,7,2,3,4,5,6] => ? = 1 + 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,1,2,8] => [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,1,3,4] => [6,1,2,8,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7] => ? = 5 + 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
 => [7,8,6,2,3,1,4,5] => [6,1,2,3,8,7,4,5] => [6,1,2,3,7,4,5] => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [5,6,7,2,3,1,4,8] => [6,1,2,7,3,4,5,8] => [6,1,2,7,3,4,5] => ? = 2 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => [6,1,2,3,4,5,8,7] => [6,1,2,3,4,5,7] => ? = 1 + 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7,8] => [6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7] => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,1,2,4,6] => [1,6,2,7,3,8,4,5] => [1,6,2,7,3,4,5] => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
 => [5,6,7,3,1,2,4,8] => [1,6,2,7,3,4,5,8] => [1,6,2,7,3,4,5] => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
 => [7,8,6,2,1,3,4,5] => [5,1,2,3,8,7,4,6] => [5,1,2,3,7,4,6] => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [8,6,7,2,1,3,4,5] => [5,1,2,3,8,4,7,6] => [5,1,2,3,4,7,6] => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
 => [6,7,8,2,1,3,4,5] => [5,1,2,3,8,4,6,7] => [5,1,2,3,4,6,7] => ? = 1 + 1

Description

The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.

Matching statistic: St001124
(load all 2 compositions to match this statistic)

Mapping

Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 84%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => [1,0]
 => []
 => ?
 => ? = 0 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [1]
 => []
 => ? = 0 - 1
[1,1,0,0]
 => [1,1,0,0]
 => []
 => ?
 => ? = 0 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => []
 => ? = 0 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => []
 => ? = 0 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => [1]
 => ? = 1 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => []
 => ?
 => ? = 0 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => []
 => ? = 0 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => []
 => ? = 0 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => []
 => ? = 0 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => ? = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 => ? = 0 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => ? = 0 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => ? = 0 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 => ? = 0 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [4,3,2,1]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [3,2,1]
 => 2 = 3 - 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]
 => [5,3,2,1]
 => [3,2,1]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [3,2,1,1]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 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]
 => [5,4,2,1]
 => [4,2,1]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [3,2,1,1]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [3,2,2,1]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [2,1,1,1]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1]
 => [1]
 => ? = 1 - 1
[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
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 => ? = 0 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 => ? = 0 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 => ? = 0 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 => ? = 0 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1]
 => ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? = 0 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [3,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [4,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1]
 => [1]
 => ? = 1 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1]
 => []
 => ? = 0 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => []
 => ? = 0 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [3]
 => []
 => ? = 0 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => []
 => ? = 0 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [5]
 => []
 => ? = 0 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => []
 => ? = 0 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => [1]
 => ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => ? = 6 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [2,1]
 => [1]
 => ? = 1 - 1

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

Matching statistic: St000829

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 84%●distinct values known / distinct values provided: 60%

Values

[1,0]
 => [1] => [1] => [] => ? = 0
[1,0,1,0]
 => [2,1] => [2,1] => [1] => ? = 0
[1,1,0,0]
 => [1,2] => [1,2] => [1] => ? = 0
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [2,1] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,2] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [3,2,1] => 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,1,2] => [3,1,2] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,1,3,2] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,4,2,3] => [1,2,3] => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1,4,2] => [3,1,2] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => [3,1,2] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,4,3,1,2] => [4,3,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,4,1,3,2] => [4,1,3,2] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,2,1,3] => [4,2,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,1,4,3,2] => [1,4,3,2] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,1,4,2,3] => [1,4,2,3] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,1,4,2] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,1,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,5,4,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,5,2,4,3] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,5,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,5,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1,5,3,2] => [4,1,3,2] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,1,5,2,3] => [4,1,2,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,1,5,2] => [4,3,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,3,1,2,5] => [4,3,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [4,1,2,5,3] => [4,1,2,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,2,5,3,4] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,4,2,5,3] => [1,4,2,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ? = 2
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,8,2,1] => [8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => ? = 2
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [4,5,6,3,7,8,2,1] => [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,7,8,2,1] => [8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => ? = 3
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,8,2,1] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,2,3,1] => [8,1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 5
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,4,3,2,6,7,8,1] => [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 5
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,2,1,3] => [7,1,8,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,2,1,4] => [7,6,1,8,2,3,4,5] => [7,6,1,2,3,4,5] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,1,8] => [7,6,5,4,3,1,2,8] => [7,6,5,4,3,1,2] => ? = 5
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,3,2,1,8] => [7,6,5,4,1,2,3,8] => [7,6,5,4,1,2,3] => ? = 4
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,2,1,8] => [7,6,5,1,2,3,4,8] => [7,6,5,1,2,3,4] => ? = 3
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,2,1,7] => [7,6,1,2,3,4,8,5] => [7,6,1,2,3,4,5] => ? = 2
[1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,3,1,4] => [7,1,2,8,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,2,4,1,5] => [7,5,1,2,8,3,4,6] => [7,5,1,2,3,4,6] => ? = 2
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,1,8] => [7,5,1,2,3,4,6,8] => [7,5,1,2,3,4,6] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [7,8,2,3,4,5,1,6] => [7,1,2,3,4,8,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,6,1,7] => [7,3,1,2,4,5,8,6] => [7,3,1,2,4,5,6] => ? = 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,1,8] => [7,3,1,2,4,5,6,8] => [7,3,1,2,4,5,6] => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,6,1,7] => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,5,6] => ? = 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,1,8] => [7,2,1,3,4,5,6,8] => [7,2,1,3,4,5,6] => ? = 2
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1,8] => [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,1,2,4] => [1,7,2,8,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,1,2,5] => [1,7,2,3,8,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
 => [7,8,3,4,5,1,2,6] => [1,7,2,3,4,8,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,1,2,7] => [1,7,2,3,4,5,8,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,1,2,8] => [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,1,3,4] => [6,1,2,8,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7] => ? = 5
[1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
 => [7,8,6,2,3,1,4,5] => [6,1,2,3,8,7,4,5] => [6,1,2,3,7,4,5] => ? = 2
[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [5,6,7,2,3,1,4,8] => [6,1,2,7,3,4,5,8] => [6,1,2,7,3,4,5] => ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => [6,1,2,3,4,5,8,7] => [6,1,2,3,4,5,7] => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7,8] => [6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,1,2,4,6] => [1,6,2,7,3,8,4,5] => [1,6,2,7,3,4,5] => ? = 2
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
 => [5,6,7,3,1,2,4,8] => [1,6,2,7,3,4,5,8] => [1,6,2,7,3,4,5] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
 => [7,8,6,2,1,3,4,5] => [5,1,2,3,8,7,4,6] => [5,1,2,3,7,4,6] => ? = 2

Description

The Ulam distance of a permutation to the identity permutation. This is, for a permutation

$\pi$ of

$n$ , given by

$n$ minus the length of the longest increasing subsequence of

$\pi^{-1}$ . In other words, this statistic plus [[St000062]] equals

$n$ .

The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001298The number of repeated entries in the Lehmer code of a permutation. St000021The number of descents of a permutation. St000204The number of internal nodes of a binary tree. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000312The number of leaves in a graph. St000711The number of big exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000871The number of very big ascents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000445The number of rises of length 1 of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001427The number of descents of a signed permutation. St000015The number of peaks of a Dyck path. St000236The number of cyclical small weak excedances. St000443The number of long tunnels of a Dyck path. St000824The sum of the number of descents and the number of recoils of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001368The number of vertices of maximal degree in a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset. St000731The number of double exceedences of a permutation. St000834The number of right outer peaks of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001520The number of strict 3-descents. St001570The minimal number of edges to add to make a graph Hamiltonian.