- FindStat

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

Matching statistic: St000369
(load all 4 compositions to match this statistic)

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv deficit of a Dyck path. For a Dyck path

$D$ of semilength

$n$ , this is defined as

$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$ In other words, this is the number of boxes in the partition traced out by

$D$ for which the leg-length minus the arm-length is not in

$\{0,1\}$ . See also [[St000376]] for the bounce deficit.

Matching statistic: St000376
(load all 2 compositions to match this statistic)

Mapping

St000376: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => 3
[1,1,1,0,0,1,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 9
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 8
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 9
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 7
[1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 7
[1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => ? = 6
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 6
[1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 9
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 12
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 7
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 6
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 11
[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5
[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 11
[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 5
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 10
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 10
[1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 7
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => ? = 8
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 5
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 5
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 7
[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 8
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 9
[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 8
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => ? = 9
[1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => ? = 7
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 7
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 6
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 6
[1,1,0,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 7
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 5
[1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 5
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => ? = 5
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 6
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 7
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 9
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1

Description

The bounce deficit of a Dyck path. For a Dyck path

$D$ of semilength

$n$ , this is defined as

$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$ The zeta map [[Mp00032]] sends this statistic to the dinv deficit [[St000369]], both are thus equidistributed.

Matching statistic: St000377

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 79%

Values

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

Description

The dinv defect of an integer partition. This is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$ .