- FindStat

Your data matches 33 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000013

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000392
(load all 6 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 97%●distinct values known / distinct values provided: 90%

Values

[1] => [1] => 1 => 1 => 1 = 2 - 1
[1,1] => [2] => 10 => 01 => 1 = 2 - 1
[2] => [1,1] => 11 => 11 => 2 = 3 - 1
[1,1,1] => [3] => 100 => 001 => 1 = 2 - 1
[1,2] => [1,2] => 110 => 011 => 2 = 3 - 1
[2,1] => [2,1] => 101 => 101 => 1 = 2 - 1
[3] => [1,1,1] => 111 => 111 => 3 = 4 - 1
[1,1,1,1] => [4] => 1000 => 0001 => 1 = 2 - 1
[1,1,2] => [1,3] => 1100 => 0011 => 2 = 3 - 1
[1,2,1] => [2,2] => 1010 => 0101 => 1 = 2 - 1
[1,3] => [1,1,2] => 1110 => 0111 => 3 = 4 - 1
[2,1,1] => [3,1] => 1001 => 1001 => 1 = 2 - 1
[2,2] => [1,2,1] => 1101 => 1011 => 2 = 3 - 1
[3,1] => [2,1,1] => 1011 => 1101 => 2 = 3 - 1
[4] => [1,1,1,1] => 1111 => 1111 => 4 = 5 - 1
[1,1,1,1,1] => [5] => 10000 => 00001 => 1 = 2 - 1
[1,1,1,2] => [1,4] => 11000 => 00011 => 2 = 3 - 1
[1,1,2,1] => [2,3] => 10100 => 00101 => 1 = 2 - 1
[1,1,3] => [1,1,3] => 11100 => 00111 => 3 = 4 - 1
[1,2,1,1] => [3,2] => 10010 => 01001 => 1 = 2 - 1
[1,2,2] => [1,2,2] => 11010 => 01011 => 2 = 3 - 1
[1,3,1] => [2,1,2] => 10110 => 01101 => 2 = 3 - 1
[1,4] => [1,1,1,2] => 11110 => 01111 => 4 = 5 - 1
[2,1,1,1] => [4,1] => 10001 => 10001 => 1 = 2 - 1
[2,1,2] => [1,3,1] => 11001 => 10011 => 2 = 3 - 1
[2,2,1] => [2,2,1] => 10101 => 10101 => 1 = 2 - 1
[2,3] => [1,1,2,1] => 11101 => 10111 => 3 = 4 - 1
[3,1,1] => [3,1,1] => 10011 => 11001 => 2 = 3 - 1
[3,2] => [1,2,1,1] => 11011 => 11011 => 2 = 3 - 1
[4,1] => [2,1,1,1] => 10111 => 11101 => 3 = 4 - 1
[5] => [1,1,1,1,1] => 11111 => 11111 => 5 = 6 - 1
[1,1,1,1,1,1] => [6] => 100000 => 000001 => 1 = 2 - 1
[1,1,1,1,2] => [1,5] => 110000 => 000011 => 2 = 3 - 1
[1,1,1,2,1] => [2,4] => 101000 => 000101 => 1 = 2 - 1
[1,1,1,3] => [1,1,4] => 111000 => 000111 => 3 = 4 - 1
[1,1,2,1,1] => [3,3] => 100100 => 001001 => 1 = 2 - 1
[1,1,2,2] => [1,2,3] => 110100 => 001011 => 2 = 3 - 1
[1,1,3,1] => [2,1,3] => 101100 => 001101 => 2 = 3 - 1
[1,1,4] => [1,1,1,3] => 111100 => 001111 => 4 = 5 - 1
[1,2,1,1,1] => [4,2] => 100010 => 010001 => 1 = 2 - 1
[1,2,1,2] => [1,3,2] => 110010 => 010011 => 2 = 3 - 1
[1,2,2,1] => [2,2,2] => 101010 => 010101 => 1 = 2 - 1
[1,2,3] => [1,1,2,2] => 111010 => 010111 => 3 = 4 - 1
[1,3,1,1] => [3,1,2] => 100110 => 011001 => 2 = 3 - 1
[1,3,2] => [1,2,1,2] => 110110 => 011011 => 2 = 3 - 1
[1,4,1] => [2,1,1,2] => 101110 => 011101 => 3 = 4 - 1
[1,5] => [1,1,1,1,2] => 111110 => 011111 => 5 = 6 - 1
[2,1,1,1,1] => [5,1] => 100001 => 100001 => 1 = 2 - 1
[2,1,1,2] => [1,4,1] => 110001 => 100011 => 2 = 3 - 1
[2,1,2,1] => [2,3,1] => 101001 => 100101 => 1 = 2 - 1
[2,8] => [1,1,1,1,1,1,1,2,1] => 1111111101 => 1011111111 => ? = 9 - 1
[3,7] => [1,1,1,1,1,1,2,1,1] => 1111111011 => 1101111111 => ? = 8 - 1
[5,5] => [1,1,1,1,2,1,1,1,1] => 1111101111 => 1111011111 => ? = 6 - 1
[10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 11 - 1

Description

The length of the longest run of ones in a binary word.

Matching statistic: St000011
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 1 = 2 - 1
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 2 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 2 = 3 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,1,3] => [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]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4 - 1
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6 - 1
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => ? = 4 - 1
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 4 - 1
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]
 => ? = 4 - 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 4 - 1
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => ? = 4 - 1
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5 - 1
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 2 - 1
[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 - 1
[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 7 - 1
[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6 - 1
[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9 - 1
[3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 8 - 1
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10 - 1

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000439

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 2
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 3
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 4
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 5
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 4
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 3
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 4
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 4
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
 => ? = 6
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 4
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 4
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 4
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 5
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? = 8
[2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
 => ? = 8
[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 7
[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 6
[5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 7
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2
[2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
 => ? = 9
[3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 8
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 6
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 9
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 10

Description

The position of the first down step of a Dyck path.

Matching statistic: St001933

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [[1],[]]
 => [1]
 => 1 = 2 - 1
[1,1] => [2] => [[2],[]]
 => [2]
 => 1 = 2 - 1
[2] => [1,1] => [[1,1],[]]
 => [1,1]
 => 2 = 3 - 1
[1,1,1] => [3] => [[3],[]]
 => [3]
 => 1 = 2 - 1
[1,2] => [2,1] => [[2,2],[1]]
 => [2,2]
 => 2 = 3 - 1
[2,1] => [1,2] => [[2,1],[]]
 => [2,1]
 => 1 = 2 - 1
[3] => [1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => 3 = 4 - 1
[1,1,1,1] => [4] => [[4],[]]
 => [4]
 => 1 = 2 - 1
[1,1,2] => [3,1] => [[3,3],[2]]
 => [3,3]
 => 2 = 3 - 1
[1,2,1] => [2,2] => [[3,2],[1]]
 => [3,2]
 => 1 = 2 - 1
[1,3] => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 3 = 4 - 1
[2,1,1] => [1,3] => [[3,1],[]]
 => [3,1]
 => 1 = 2 - 1
[2,2] => [1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => 2 = 3 - 1
[3,1] => [1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 2 = 3 - 1
[4] => [1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => 4 = 5 - 1
[1,1,1,1,1] => [5] => [[5],[]]
 => [5]
 => 1 = 2 - 1
[1,1,1,2] => [4,1] => [[4,4],[3]]
 => [4,4]
 => 2 = 3 - 1
[1,1,2,1] => [3,2] => [[4,3],[2]]
 => [4,3]
 => 1 = 2 - 1
[1,1,3] => [3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => 3 = 4 - 1
[1,2,1,1] => [2,3] => [[4,2],[1]]
 => [4,2]
 => 1 = 2 - 1
[1,2,2] => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 2 = 3 - 1
[1,3,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 2 = 3 - 1
[1,4] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 4 = 5 - 1
[2,1,1,1] => [1,4] => [[4,1],[]]
 => [4,1]
 => 1 = 2 - 1
[2,1,2] => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 2 = 3 - 1
[2,2,1] => [1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 1 = 2 - 1
[2,3] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 3 = 4 - 1
[3,1,1] => [1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 2 = 3 - 1
[3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => 2 = 3 - 1
[4,1] => [1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 3 = 4 - 1
[5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => 5 = 6 - 1
[1,1,1,1,1,1] => [6] => [[6],[]]
 => [6]
 => 1 = 2 - 1
[1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => [5,5]
 => 2 = 3 - 1
[1,1,1,2,1] => [4,2] => [[5,4],[3]]
 => [5,4]
 => 1 = 2 - 1
[1,1,1,3] => [4,1,1] => [[4,4,4],[3,3]]
 => [4,4,4]
 => 3 = 4 - 1
[1,1,2,1,1] => [3,3] => [[5,3],[2]]
 => [5,3]
 => 1 = 2 - 1
[1,1,2,2] => [3,2,1] => [[4,4,3],[3,2]]
 => [4,4,3]
 => 2 = 3 - 1
[1,1,3,1] => [3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => 2 = 3 - 1
[1,1,4] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [3,3,3,3]
 => 4 = 5 - 1
[1,2,1,1,1] => [2,4] => [[5,2],[1]]
 => [5,2]
 => 1 = 2 - 1
[1,2,1,2] => [2,3,1] => [[4,4,2],[3,1]]
 => [4,4,2]
 => 2 = 3 - 1
[1,2,2,1] => [2,2,2] => [[4,3,2],[2,1]]
 => [4,3,2]
 => 1 = 2 - 1
[1,2,3] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [3,3,3,2]
 => 3 = 4 - 1
[1,3,1,1] => [2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => 2 = 3 - 1
[1,3,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [3,3,2,2]
 => 2 = 3 - 1
[1,4,1] => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => 3 = 4 - 1
[1,5] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [2,2,2,2,2]
 => 5 = 6 - 1
[2,1,1,1,1] => [1,5] => [[5,1],[]]
 => [5,1]
 => 1 = 2 - 1
[2,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => [4,4,1]
 => 2 = 3 - 1
[2,1,2,1] => [1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => 1 = 2 - 1
[1,1,1,1,1,2,1] => [6,2] => [[7,6],[5]]
 => ?
 => ? = 2 - 1
[1,1,1,1,1,3] => [6,1,1] => [[6,6,6],[5,5]]
 => [6,6,6]
 => ? = 4 - 1
[1,1,1,1,3,1] => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? = 3 - 1
[1,1,1,5] => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => [4,4,4,4,4]
 => ? = 6 - 1
[1,1,6] => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
 => [3,3,3,3,3,3]
 => ? = 7 - 1
[1,2,1,1,1,1,1] => [2,6] => [[7,2],[1]]
 => ?
 => ? = 2 - 1
[2,4,1,1] => [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
 => ?
 => ? = 4 - 1
[3,1,1,3] => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 4 - 1
[4,2,1,1] => [1,1,1,2,3] => [[4,2,1,1,1],[1]]
 => ?
 => ? = 4 - 1
[1,1,1,1,1,1,2,1] => [7,2] => [[8,7],[6]]
 => ?
 => ? = 2 - 1
[1,1,7] => [3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
 => [3,3,3,3,3,3,3]
 => ? = 8 - 1
[2,2,2,2,1] => [1,2,2,2,2] => [[5,4,3,2,1],[3,2,1]]
 => ?
 => ? = 2 - 1
[2,7] => [1,2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,1],[1,1,1,1,1,1]]
 => ?
 => ? = 8 - 1
[3,6] => [1,1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1,1],[1,1,1,1,1]]
 => ?
 => ? = 7 - 1
[4,5] => [1,1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1,1],[1,1,1,1]]
 => ?
 => ? = 6 - 1
[5,4] => [1,1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1,1],[1,1,1]]
 => ?
 => ? = 5 - 1
[6,3] => [1,1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1,1],[1,1]]
 => ?
 => ? = 6 - 1
[7,2] => [1,1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1,1],[1]]
 => ?
 => ? = 7 - 1
[1,1,1,1,1,1,1,1,2] => [9,1] => [[9,9],[8]]
 => [9,9]
 => ? = 3 - 1
[1,9] => [2,1,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
 => [2,2,2,2,2,2,2,2,2]
 => ? = 10 - 1
[2,8] => [1,2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,1],[1,1,1,1,1,1,1]]
 => ?
 => ? = 9 - 1
[3,7] => [1,1,2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,1,1],[1,1,1,1,1,1]]
 => ?
 => ? = 8 - 1
[5,5] => [1,1,1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1,1,1],[1,1,1,1]]
 => ?
 => ? = 6 - 1
[10,1] => [1,1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1,1],[]]
 => ?
 => ? = 10 - 1

Description

The largest multiplicity of a part in an integer partition.

Matching statistic: St000678

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 80%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => ? = 2 - 1
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2 = 3 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4 = 5 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5 = 6 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 3 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,2] => [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]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 3 - 1
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,1,3] => [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]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 4 - 1
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 3 - 1
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 6 - 1
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0]
 => ? = 4 - 1
[1,1,6] => [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]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 7 - 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 - 1
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [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]
 => ? = 8 - 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
 => ? = 4 - 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 8 - 1
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 9 - 1
[2,1,1,1,1,1,1,1] => [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]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 8 - 1
[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 7 - 1
[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5 - 1
[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 7 - 1
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 - 1
[9] => [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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,2] => [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]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ? = 3 - 1
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 10 - 1
[2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 9 - 1
[3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 8 - 1
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 9 - 1
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 11 - 1
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 10 - 1

Description

The number of up steps after the last double rise of a Dyck path.

Matching statistic: St001038

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 80%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => ? = 2 - 1
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3 = 4 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 5 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 3 = 4 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3 = 4 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5 = 6 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,2] => [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]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 - 1
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,1,3] => [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]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 4 - 1
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => ? = 3 - 1
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 4 - 1
[1,1,6] => [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]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 7 - 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 6 - 1
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 8 - 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => ? = 4 - 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 8 - 1
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 9 - 1
[2,1,1,1,1,1,1,1] => [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]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 8 - 1
[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 7 - 1
[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5 - 1
[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 6 - 1
[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 7 - 1
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 8 - 1
[9] => [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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,2] => [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]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 - 1
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 10 - 1
[2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 9 - 1
[3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 8 - 1
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 9 - 1
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 11 - 1
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 10 - 1

Description

The minimal height of a column in the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000700

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 70% ●values known / values provided: 72%●distinct values known / distinct values provided: 70%

Values

[1] => [1,0]
 => [1,0]
 => [[]]
 => 1 = 2 - 1
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [[],[]]
 => 1 = 2 - 1
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [[[]]]
 => 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[[]],[]]
 => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 2 = 3 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => 2 = 3 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[[],[],[]]]]
 => 3 = 4 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[[],[]]]]]
 => 4 = 5 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[[[]],[[]]]]
 => 3 = 4 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[[]],[],[]]]
 => 2 = 3 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[[]]],[[]]]
 => 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[[[]],[]]]]
 => 3 = 4 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[]]
 => 1 = 2 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[]]]
 => 2 = 3 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[[],[],[],[]],[]]
 => 1 = 2 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[[[],[],[],[]]]]
 => 3 = 4 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[[],[],[]],[],[]]
 => 1 = 2 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [[[]],[[],[],[]]]
 => 2 = 3 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[[[],[],[]],[]]]
 => 2 = 3 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [[[[[],[],[]]]]]
 => 4 = 5 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[[],[]],[],[],[]]
 => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[[],[]],[[],[]]]
 => 2 = 3 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[[]],[[],[]],[]]
 => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[[[]],[[],[]]]]
 => 3 = 4 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[[[],[]],[],[]]]
 => 2 = 3 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [[[[],[]]],[[]]]
 => 2 = 3 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [[[[[],[]],[]]]]
 => 3 = 4 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[[[[[],[]]]]]]
 => 5 = 6 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[]],[],[],[],[]]
 => 1 = 2 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[[],[],[]],[[]]]
 => 2 = 3 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[[],[]],[[]],[]]
 => 1 = 2 - 1
[1,1,1,1,1,1,2] => [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]
 => [[[],[],[],[],[],[],[]]]
 => ? = 3 - 1
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [[[],[],[],[],[],[]],[]]
 => ? = 2 - 1
[1,1,1,1,1,3] => [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 - 1
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [[[[],[],[],[],[]],[]]]
 => ? = 3 - 1
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [[[[[[],[],[],[]]]]]]
 => ? = 6 - 1
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
 => [[[[[],[],[]]],[[]]]]
 => ? = 4 - 1
[1,1,6] => [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]
 => [[[[[[[],[],[]]]]]]]
 => ? = 7 - 1
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => [[[[]],[[]],[[],[]]]]
 => ? = 4 - 1
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
 => [[[[[[]],[[],[]]]]]]
 => ? = 6 - 1
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [[[[[[[],[]],[]]]]]]
 => ? = 6 - 1
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[[],[]]]]]]]]
 => ? = 8 - 1
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
 => [[[[[]],[[]],[],[]]]]
 => ? = 4 - 1
[2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [[[[[[[]],[[]]]]]]]
 => ? = 7 - 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [[[[],[],[]]],[[[]]]]
 => ? = 4 - 1
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [[[[]],[]],[[]],[[]]]
 => ? = 3 - 1
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [[[[[[]]],[[[]]]]]]
 => ? = 6 - 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
 => [[[[[]],[],[]],[[]]]]
 => ? = 4 - 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [[[[[]]]],[[[[]]]]]
 => ? = 5 - 1
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => [[[[[[]]]],[[[]]]]]
 => ? = 5 - 1
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => [[[[[[[]]],[[]]]]]]
 => ? = 6 - 1
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[[[[[[[]],[]]]]]]]
 => ? = 7 - 1
[8] => [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]
 => [[[[[[[[[]]]]]]]]]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[],[],[],[]]
 => ? = 2 - 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [[[],[],[],[],[],[],[]],[]]
 => ? = 2 - 1
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[[],[],[]]]]]]]]
 => ? = 8 - 1
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [[[[[[[[[],[]]]]]]]]]
 => ? = 9 - 1
[2,1,1,1,1,1,1,1] => [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]
 => [[[]],[],[],[],[],[],[],[]]
 => ? = 2 - 1
[2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[[]],[[]],[]]
 => ? = 2 - 1
[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
 => [[[[[[[[]],[[]]]]]]]]
 => ? = 8 - 1
[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [[[[[[[]]],[[[]]]]]]]
 => ? = 7 - 1
[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]],[[[[]]]]]]
 => ? = 6 - 1
[5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [[[[[[]]]]],[[[[]]]]]
 => ? = 5 - 1
[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
 => [[[[[[[]]]],[[[]]]]]]
 => ? = 6 - 1
[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
 => [[[[[[[[]]],[[]]]]]]]
 => ? = 7 - 1
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [[[[[[[[[]],[]]]]]]]]
 => ? = 8 - 1
[9] => [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]
 => [[[[[[[[[[]]]]]]]]]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[],[],[],[],[]]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,2] => [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]
 => [[[],[],[],[],[],[],[],[],[]]]
 => ? = 3 - 1
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [[[[[[[[[[],[]]]]]]]]]]
 => ? = 10 - 1
[2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
 => [[[[[[[[[]],[[]]]]]]]]]
 => ? = 9 - 1
[3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
 => [[[[[[[[]]],[[[]]]]]]]]
 => ? = 8 - 1
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]],[[[[[]]]]]]
 => ? = 6 - 1
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [[[[[[[[[[]],[]]]]]]]]]
 => ? = 9 - 1
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [[[[[[[[[[[]]]]]]]]]]]
 => ? = 11 - 1
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [[[[[[[[[[[]],[]]]]]]]]]]
 => ? = 10 - 1

Description

The protection number of an ordered tree. This is the minimal distance from the root to a leaf.

Matching statistic: St000969
(load all 5 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 70%

Values

[1] => [1] => [1,0]
 => 2
[1,1] => [2] => [1,1,0,0]
 => 2
[2] => [1,1] => [1,0,1,0]
 => 3
[1,1,1] => [3] => [1,1,1,0,0,0]
 => 2
[1,2] => [1,2] => [1,0,1,1,0,0]
 => 3
[2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[3] => [1,1,1] => [1,0,1,0,1,0]
 => 4
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 2
[1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
[2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 2
[1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 4
[1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5
[2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 2
[1,1,1,1,2] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 3
[1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
[1,1,2,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
[1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
[1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 4
[1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
[1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
[1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 6
[2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 3
[2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2
[1,1,1,1,1,1,2] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 3
[1,1,1,1,1,2,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[1,1,1,1,1,3] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4
[1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[1,1,1,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 6
[1,1,4,2] => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[1,1,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7
[1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[1,2,2,2,1] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[1,2,2,3] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[1,2,5] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6
[1,6,1] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[2,1,1,1,1,1,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
[2,4,1,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[2,6] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[3,1,1,3] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 4
[3,2,2,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
[3,5] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6
[4,2,1,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[4,4] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5
[5,3] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[6,2] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 2
[1,1,1,1,1,1,2,1] => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[1,1,7] => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 8
[1,8] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 9
[2,1,1,1,1,1,1,1] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
[2,2,2,2,1] => [2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[2,7] => [1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8
[3,6] => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 7
[4,5] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6
[5,4] => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[6,3] => [1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,2] => [1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[9] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,1,1,1,1,1,1,1,1] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1,2] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 3
[1,9] => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 10
[2,8] => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 9
[3,7] => [1,1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 8
[5,5] => [1,1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[9,1] => [2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 11
[10,1] => [2,1,1,1,1,1,1,1,1,1] => ?
 => ? = 10

Description

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . Then we calculate the global dimension of that CNakayama algebra.

Matching statistic: St000723
(load all 8 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000723: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 70%

Values

[1] => [1] => ([],1)
 => 1 = 2 - 1
[1,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[2] => [2] => ([],2)
 => 2 = 3 - 1
[1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[2,1] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[3] => [3] => ([],3)
 => 3 = 4 - 1
[1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,1] => [1,3] => ([(2,3)],4)
 => 2 = 3 - 1
[4] => [4] => ([],4)
 => 4 = 5 - 1
[1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[4,1] => [1,4] => ([(3,4)],5)
 => 3 = 4 - 1
[5] => [5] => ([],5)
 => 5 = 6 - 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,1,1,2] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,1,3] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,2,2] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,1,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,2,1,1,1] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 6 - 1
[2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[2,1,1,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 - 1
[1,1,1,1,1,2,1] => [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,1,1,1,1,3] => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[1,1,1,1,3,1] => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 - 1
[1,1,1,5] => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 - 1
[1,1,4,2] => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[1,1,6] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 - 1
[1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,2,2,2,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,2,2,3] => [3,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[1,2,5] => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 - 1
[1,6,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 - 1
[1,7] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 8 - 1
[2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,4,1,1] => [1,1,4,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 7 - 1
[3,1,1,3] => [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[3,2,2,1] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 - 1
[3,5] => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 - 1
[4,2,1,1] => [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[4,4] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 5 - 1
[5,3] => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 5 - 1
[6,2] => [2,6] => ([(5,7),(6,7)],8)
 => ? = 6 - 1
[7,1] => [1,7] => ([(6,7)],8)
 => ? = 7 - 1
[8] => [8] => ([],8)
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 - 1
[1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 - 1
[1,1,7] => [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 8 - 1
[1,8] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 9 - 1
[2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 - 1
[2,2,2,2,1] => [1,2,2,2,2] => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 - 1
[2,7] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 8 - 1
[3,6] => [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 7 - 1
[4,5] => [5,4] => ([(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 6 - 1
[5,4] => [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 5 - 1
[6,3] => [3,6] => ([(5,8),(6,8),(7,8)],9)
 => ? = 6 - 1
[7,2] => [2,7] => ([(6,8),(7,8)],9)
 => ? = 7 - 1
[8,1] => [1,8] => ([(7,8)],9)
 => ? = 8 - 1
[9] => [9] => ([],9)
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2 - 1
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3 - 1
[1,9] => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 10 - 1
[2,8] => [8,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 9 - 1
[3,7] => [7,3] => ([(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 8 - 1
[5,5] => [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 6 - 1
[9,1] => [1,9] => ([(8,9)],10)
 => ? = 9 - 1
[10] => [10] => ([],10)
 => ? = 11 - 1
[10,1] => [1,10] => ([(9,10)],11)
 => ? = 10 - 1

Description

The maximal cardinality of a set of vertices with the same neighbourhood in a graph. The set of so called mating graphs, for which this statistic equals

$1$ , is enumerated by [1].

The following 23 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000025The number of initial rises of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000007The number of saliances of the permutation. St000654The first descent of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000989The number of final rises of a permutation. St001062The maximal size of a block of a set partition. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.