Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001460
Mapping
St001460: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => 1
[[1],[1]]
 => 1
[[2]]
 => 1
[[1,1]]
 => 2
[[1],[1],[1]]
 => 1
[[2],[1]]
 => 1
[[1,1],[1]]
 => 2
[[3]]
 => 1
[[2,1]]
 => 2
[[1,1,1]]
 => 3
[[1],[1],[1],[1]]
 => 1
[[2],[1],[1]]
 => 1
[[2],[2]]
 => 1
[[1,1],[1],[1]]
 => 2
[[1,1],[1,1]]
 => 2
[[3],[1]]
 => 1
[[2,1],[1]]
 => 2
[[1,1,1],[1]]
 => 3
[[4]]
 => 1
[[3,1]]
 => 2
[[2,2]]
 => 2
[[2,1,1]]
 => 3
[[1,1,1,1]]
 => 4
[[1],[1],[1],[1],[1]]
 => 1
[[2],[1],[1],[1]]
 => 1
[[2],[2],[1]]
 => 1
[[1,1],[1],[1],[1]]
 => 2
[[1,1],[1,1],[1]]
 => 2
[[3],[1],[1]]
 => 1
[[3],[2]]
 => 1
[[2,1],[1],[1]]
 => 2
[[2,1],[2]]
 => 2
[[2,1],[1,1]]
 => 2
[[1,1,1],[1],[1]]
 => 3
[[1,1,1],[1,1]]
 => 3
[[4],[1]]
 => 1
[[3,1],[1]]
 => 2
[[2,2],[1]]
 => 2
[[2,1,1],[1]]
 => 3
[[1,1,1,1],[1]]
 => 4
[[5]]
 => 1
[[4,1]]
 => 2
[[3,2]]
 => 2
[[3,1,1]]
 => 3
[[2,2,1]]
 => 3
[[2,1,1,1]]
 => 4
[[1,1,1,1,1]]
 => 5
[[1],[1],[1],[1],[1],[1]]
 => 1
[[2],[1],[1],[1],[1]]
 => 1
[[2],[2],[1],[1]]
 => 1
Description
Number of columns of a plane partition.
Matching statistic: St001446
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
St001446: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => 1
[[1],[1]]
 => [[1,1]]
 => 1
[[2]]
 => [[2]]
 => 1
[[1,1]]
 => [[1],[1]]
 => 2
[[1],[1],[1]]
 => [[1,1,1]]
 => 1
[[2],[1]]
 => [[2,1]]
 => 1
[[1,1],[1]]
 => [[1,1],[1]]
 => 2
[[3]]
 => [[3]]
 => 1
[[2,1]]
 => [[2],[1]]
 => 2
[[1,1,1]]
 => [[1],[1],[1]]
 => 3
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => 1
[[2],[1],[1]]
 => [[2,1,1]]
 => 1
[[2],[2]]
 => [[2,2]]
 => 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => 2
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => 2
[[3],[1]]
 => [[3,1]]
 => 1
[[2,1],[1]]
 => [[2,1],[1]]
 => 2
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => 3
[[4]]
 => [[4]]
 => 1
[[3,1]]
 => [[3],[1]]
 => 2
[[2,2]]
 => [[2],[2]]
 => 2
[[2,1,1]]
 => [[2],[1],[1]]
 => 3
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => 4
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => 1
[[2],[2],[1]]
 => [[2,2,1]]
 => 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => 2
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => 2
[[3],[1],[1]]
 => [[3,1,1]]
 => 1
[[3],[2]]
 => [[3,2]]
 => 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => 2
[[2,1],[2]]
 => [[2,2],[1]]
 => 2
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => 2
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => 3
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => 3
[[4],[1]]
 => [[4,1]]
 => 1
[[3,1],[1]]
 => [[3,1],[1]]
 => 2
[[2,2],[1]]
 => [[2,1],[2]]
 => 2
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => 3
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => 4
[[5]]
 => [[5]]
 => 1
[[4,1]]
 => [[4],[1]]
 => 2
[[3,2]]
 => [[3],[2]]
 => 2
[[3,1,1]]
 => [[3],[1],[1]]
 => 3
[[2,2,1]]
 => [[2],[2],[1]]
 => 3
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => 4
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => 5
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => 1
Description
Number of rows in the plane partition.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => 1
[[1],[1]]
 => [[1,1]]
 => [2]
 => 1
[[2]]
 => [[2]]
 => [2]
 => 1
[[1,1]]
 => [[1],[1]]
 => [1,1]
 => 2
[[1],[1],[1]]
 => [[1,1,1]]
 => [3]
 => 1
[[2],[1]]
 => [[2,1]]
 => [3]
 => 1
[[1,1],[1]]
 => [[1,1],[1]]
 => [2,1]
 => 2
[[3]]
 => [[3]]
 => [3]
 => 1
[[2,1]]
 => [[2],[1]]
 => [2,1]
 => 2
[[1,1,1]]
 => [[1],[1],[1]]
 => [1,1,1]
 => 3
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => [4]
 => 1
[[2],[1],[1]]
 => [[2,1,1]]
 => [4]
 => 1
[[2],[2]]
 => [[2,2]]
 => [4]
 => 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => [3,1]
 => 2
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => [2,2]
 => 2
[[3],[1]]
 => [[3,1]]
 => [4]
 => 1
[[2,1],[1]]
 => [[2,1],[1]]
 => [3,1]
 => 2
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => [2,1,1]
 => 3
[[4]]
 => [[4]]
 => [4]
 => 1
[[3,1]]
 => [[3],[1]]
 => [3,1]
 => 2
[[2,2]]
 => [[2],[2]]
 => [2,2]
 => 2
[[2,1,1]]
 => [[2],[1],[1]]
 => [2,1,1]
 => 3
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => [1,1,1,1]
 => 4
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => [5]
 => 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => [5]
 => 1
[[2],[2],[1]]
 => [[2,2,1]]
 => [5]
 => 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => [4,1]
 => 2
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => [3,2]
 => 2
[[3],[1],[1]]
 => [[3,1,1]]
 => [5]
 => 1
[[3],[2]]
 => [[3,2]]
 => [5]
 => 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => [4,1]
 => 2
[[2,1],[2]]
 => [[2,2],[1]]
 => [4,1]
 => 2
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => [3,2]
 => 2
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => [3,1,1]
 => 3
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => [2,2,1]
 => 3
[[4],[1]]
 => [[4,1]]
 => [5]
 => 1
[[3,1],[1]]
 => [[3,1],[1]]
 => [4,1]
 => 2
[[2,2],[1]]
 => [[2,1],[2]]
 => [3,2]
 => 2
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => [3,1,1]
 => 3
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => 4
[[5]]
 => [[5]]
 => [5]
 => 1
[[4,1]]
 => [[4],[1]]
 => [4,1]
 => 2
[[3,2]]
 => [[3],[2]]
 => [3,2]
 => 2
[[3,1,1]]
 => [[3],[1],[1]]
 => [3,1,1]
 => 3
[[2,2,1]]
 => [[2],[2],[1]]
 => [2,2,1]
 => 3
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => [2,1,1,1]
 => 4
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => 5
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => [6]
 => 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => [6]
 => 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => [6]
 => 1
Description
The length of the partition.
Matching statistic: St000147
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => [1]
 => 1
[[1],[1]]
 => [[1,1]]
 => [2]
 => [1,1]
 => 1
[[2]]
 => [[2]]
 => [2]
 => [1,1]
 => 1
[[1,1]]
 => [[1],[1]]
 => [1,1]
 => [2]
 => 2
[[1],[1],[1]]
 => [[1,1,1]]
 => [3]
 => [1,1,1]
 => 1
[[2],[1]]
 => [[2,1]]
 => [3]
 => [1,1,1]
 => 1
[[1,1],[1]]
 => [[1,1],[1]]
 => [2,1]
 => [2,1]
 => 2
[[3]]
 => [[3]]
 => [3]
 => [1,1,1]
 => 1
[[2,1]]
 => [[2],[1]]
 => [2,1]
 => [2,1]
 => 2
[[1,1,1]]
 => [[1],[1],[1]]
 => [1,1,1]
 => [3]
 => 3
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => [4]
 => [1,1,1,1]
 => 1
[[2],[1],[1]]
 => [[2,1,1]]
 => [4]
 => [1,1,1,1]
 => 1
[[2],[2]]
 => [[2,2]]
 => [4]
 => [1,1,1,1]
 => 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => [2,2]
 => [2,2]
 => 2
[[3],[1]]
 => [[3,1]]
 => [4]
 => [1,1,1,1]
 => 1
[[2,1],[1]]
 => [[2,1],[1]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => [2,1,1]
 => [3,1]
 => 3
[[4]]
 => [[4]]
 => [4]
 => [1,1,1,1]
 => 1
[[3,1]]
 => [[3],[1]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2]]
 => [[2],[2]]
 => [2,2]
 => [2,2]
 => 2
[[2,1,1]]
 => [[2],[1],[1]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => [1,1,1,1]
 => [4]
 => 4
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[2],[2],[1]]
 => [[2,2,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => 2
[[3],[1],[1]]
 => [[3,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[3],[2]]
 => [[3,2]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[2,1],[2]]
 => [[2,2],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => 2
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 3
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => [2,2,1]
 => [3,2]
 => 3
[[4],[1]]
 => [[4,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[3,1],[1]]
 => [[3,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[2,2],[1]]
 => [[2,1],[2]]
 => [3,2]
 => [2,2,1]
 => 2
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 3
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => 4
[[5]]
 => [[5]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[4,1]]
 => [[4],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[3,2]]
 => [[3],[2]]
 => [3,2]
 => [2,2,1]
 => 2
[[3,1,1]]
 => [[3],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 3
[[2,2,1]]
 => [[2],[2],[1]]
 => [2,2,1]
 => [3,2]
 => 3
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => 4
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [5]
 => 5
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => [6]
 => [1,1,1,1,1,1]
 => 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => [6]
 => [1,1,1,1,1,1]
 => 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => [6]
 => [1,1,1,1,1,1]
 => 1
Description
The largest part of an integer partition.
Matching statistic: St000288
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => 10 => 1
[[1],[1]]
 => [[1,1]]
 => [2]
 => 100 => 1
[[2]]
 => [[2]]
 => [2]
 => 100 => 1
[[1,1]]
 => [[1],[1]]
 => [1,1]
 => 110 => 2
[[1],[1],[1]]
 => [[1,1,1]]
 => [3]
 => 1000 => 1
[[2],[1]]
 => [[2,1]]
 => [3]
 => 1000 => 1
[[1,1],[1]]
 => [[1,1],[1]]
 => [2,1]
 => 1010 => 2
[[3]]
 => [[3]]
 => [3]
 => 1000 => 1
[[2,1]]
 => [[2],[1]]
 => [2,1]
 => 1010 => 2
[[1,1,1]]
 => [[1],[1],[1]]
 => [1,1,1]
 => 1110 => 3
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => [4]
 => 10000 => 1
[[2],[1],[1]]
 => [[2,1,1]]
 => [4]
 => 10000 => 1
[[2],[2]]
 => [[2,2]]
 => [4]
 => 10000 => 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => [3,1]
 => 10010 => 2
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => [2,2]
 => 1100 => 2
[[3],[1]]
 => [[3,1]]
 => [4]
 => 10000 => 1
[[2,1],[1]]
 => [[2,1],[1]]
 => [3,1]
 => 10010 => 2
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => [2,1,1]
 => 10110 => 3
[[4]]
 => [[4]]
 => [4]
 => 10000 => 1
[[3,1]]
 => [[3],[1]]
 => [3,1]
 => 10010 => 2
[[2,2]]
 => [[2],[2]]
 => [2,2]
 => 1100 => 2
[[2,1,1]]
 => [[2],[1],[1]]
 => [2,1,1]
 => 10110 => 3
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => [1,1,1,1]
 => 11110 => 4
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => [5]
 => 100000 => 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => [5]
 => 100000 => 1
[[2],[2],[1]]
 => [[2,2,1]]
 => [5]
 => 100000 => 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => [4,1]
 => 100010 => 2
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => [3,2]
 => 10100 => 2
[[3],[1],[1]]
 => [[3,1,1]]
 => [5]
 => 100000 => 1
[[3],[2]]
 => [[3,2]]
 => [5]
 => 100000 => 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => [4,1]
 => 100010 => 2
[[2,1],[2]]
 => [[2,2],[1]]
 => [4,1]
 => 100010 => 2
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => [3,2]
 => 10100 => 2
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => [3,1,1]
 => 100110 => 3
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => [2,2,1]
 => 11010 => 3
[[4],[1]]
 => [[4,1]]
 => [5]
 => 100000 => 1
[[3,1],[1]]
 => [[3,1],[1]]
 => [4,1]
 => 100010 => 2
[[2,2],[1]]
 => [[2,1],[2]]
 => [3,2]
 => 10100 => 2
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => [3,1,1]
 => 100110 => 3
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => 101110 => 4
[[5]]
 => [[5]]
 => [5]
 => 100000 => 1
[[4,1]]
 => [[4],[1]]
 => [4,1]
 => 100010 => 2
[[3,2]]
 => [[3],[2]]
 => [3,2]
 => 10100 => 2
[[3,1,1]]
 => [[3],[1],[1]]
 => [3,1,1]
 => 100110 => 3
[[2,2,1]]
 => [[2],[2],[1]]
 => [2,2,1]
 => 11010 => 3
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => [2,1,1,1]
 => 101110 => 4
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => 111110 => 5
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => [6]
 => 1000000 => 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => [6]
 => 1000000 => 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => [6]
 => 1000000 => 1
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000378
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => [1]
 => 1
[[1],[1]]
 => [[1,1]]
 => [2]
 => [1,1]
 => 1
[[2]]
 => [[2]]
 => [2]
 => [1,1]
 => 1
[[1,1]]
 => [[1],[1]]
 => [1,1]
 => [2]
 => 2
[[1],[1],[1]]
 => [[1,1,1]]
 => [3]
 => [1,1,1]
 => 1
[[2],[1]]
 => [[2,1]]
 => [3]
 => [1,1,1]
 => 1
[[1,1],[1]]
 => [[1,1],[1]]
 => [2,1]
 => [3]
 => 2
[[3]]
 => [[3]]
 => [3]
 => [1,1,1]
 => 1
[[2,1]]
 => [[2],[1]]
 => [2,1]
 => [3]
 => 2
[[1,1,1]]
 => [[1],[1],[1]]
 => [1,1,1]
 => [2,1]
 => 3
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => [4]
 => [1,1,1,1]
 => 1
[[2],[1],[1]]
 => [[2,1,1]]
 => [4]
 => [1,1,1,1]
 => 1
[[2],[2]]
 => [[2,2]]
 => [4]
 => [1,1,1,1]
 => 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => [2,2]
 => [4]
 => 2
[[3],[1]]
 => [[3,1]]
 => [4]
 => [1,1,1,1]
 => 1
[[2,1],[1]]
 => [[2,1],[1]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => [2,1,1]
 => [2,2]
 => 3
[[4]]
 => [[4]]
 => [4]
 => [1,1,1,1]
 => 1
[[3,1]]
 => [[3],[1]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2]]
 => [[2],[2]]
 => [2,2]
 => [4]
 => 2
[[2,1,1]]
 => [[2],[1],[1]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => [1,1,1,1]
 => [3,1]
 => 4
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[2],[2],[1]]
 => [[2,2,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => [3,2]
 => [5]
 => 2
[[3],[1],[1]]
 => [[3,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[3],[2]]
 => [[3,2]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[2,1],[2]]
 => [[2,2],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => [3,2]
 => [5]
 => 2
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => [3,1,1]
 => [4,1]
 => 3
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => [2,2,1]
 => [2,2,1]
 => 3
[[4],[1]]
 => [[4,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[3,1],[1]]
 => [[3,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[2,2],[1]]
 => [[2,1],[2]]
 => [3,2]
 => [5]
 => 2
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => [3,1,1]
 => [4,1]
 => 3
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [3,1,1]
 => 4
[[5]]
 => [[5]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[4,1]]
 => [[4],[1]]
 => [4,1]
 => [2,1,1,1]
 => 2
[[3,2]]
 => [[3],[2]]
 => [3,2]
 => [5]
 => 2
[[3,1,1]]
 => [[3],[1],[1]]
 => [3,1,1]
 => [4,1]
 => 3
[[2,2,1]]
 => [[2],[2],[1]]
 => [2,2,1]
 => [2,2,1]
 => 3
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => [2,1,1,1]
 => [3,1,1]
 => 4
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [3,2]
 => 5
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => [6]
 => [1,1,1,1,1,1]
 => 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => [6]
 => [1,1,1,1,1,1]
 => 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => [6]
 => [1,1,1,1,1,1]
 => 1
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000733
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => [[1]]
 => 1
[[1],[1]]
 => [[1,1]]
 => [2]
 => [[1,2]]
 => 1
[[2]]
 => [[2]]
 => [2]
 => [[1,2]]
 => 1
[[1,1]]
 => [[1],[1]]
 => [1,1]
 => [[1],[2]]
 => 2
[[1],[1],[1]]
 => [[1,1,1]]
 => [3]
 => [[1,2,3]]
 => 1
[[2],[1]]
 => [[2,1]]
 => [3]
 => [[1,2,3]]
 => 1
[[1,1],[1]]
 => [[1,1],[1]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[3]]
 => [[3]]
 => [3]
 => [[1,2,3]]
 => 1
[[2,1]]
 => [[2],[1]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,1,1]]
 => [[1],[1],[1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => [4]
 => [[1,2,3,4]]
 => 1
[[2],[1],[1]]
 => [[2,1,1]]
 => [4]
 => [[1,2,3,4]]
 => 1
[[2],[2]]
 => [[2,2]]
 => [4]
 => [[1,2,3,4]]
 => 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[3],[1]]
 => [[3,1]]
 => [4]
 => [[1,2,3,4]]
 => 1
[[2,1],[1]]
 => [[2,1],[1]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[4]]
 => [[4]]
 => [4]
 => [[1,2,3,4]]
 => 1
[[3,1]]
 => [[3],[1]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[2,2]]
 => [[2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[2,1,1]]
 => [[2],[1],[1]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[[2],[2],[1]]
 => [[2,2,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 2
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 2
[[3],[1],[1]]
 => [[3,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[[3],[2]]
 => [[3,2]]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 2
[[2,1],[2]]
 => [[2,2],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 2
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 2
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[[4],[1]]
 => [[4,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[[3,1],[1]]
 => [[3,1],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 2
[[2,2],[1]]
 => [[2,1],[2]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 2
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[[5]]
 => [[5]]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[[4,1]]
 => [[4],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 2
[[3,2]]
 => [[3],[2]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 2
[[3,1,1]]
 => [[3],[1],[1]]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[[2,2,1]]
 => [[2],[2],[1]]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => [6]
 => [[1,2,3,4,5,6]]
 => 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => [6]
 => [[1,2,3,4,5,6]]
 => 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => [6]
 => [[1,2,3,4,5,6]]
 => 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => [[1]]
 => 0 = 1 - 1
[[1],[1]]
 => [[1,1]]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[2]]
 => [[2]]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,1]]
 => [[1],[1]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1],[1],[1]]
 => [[1,1,1]]
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
[[2],[1]]
 => [[2,1]]
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
[[1,1],[1]]
 => [[1,1],[1]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[3]]
 => [[3]]
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
[[2,1]]
 => [[2],[1]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[1,1,1]]
 => [[1],[1],[1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2],[1],[1]]
 => [[2,1,1]]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2],[2]]
 => [[2,2]]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[[3],[1]]
 => [[3,1]]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,1],[1]]
 => [[2,1],[1]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[4]]
 => [[4]]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[3,1]]
 => [[3],[1]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[2,2]]
 => [[2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[[2,1,1]]
 => [[2],[1],[1]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[[2],[2],[1]]
 => [[2,2,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 1 = 2 - 1
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 1 = 2 - 1
[[3],[1],[1]]
 => [[3,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[[3],[2]]
 => [[3,2]]
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 1 = 2 - 1
[[2,1],[2]]
 => [[2,2],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 1 = 2 - 1
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 1 = 2 - 1
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2 = 3 - 1
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2 = 3 - 1
[[4],[1]]
 => [[4,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[[3,1],[1]]
 => [[3,1],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 1 = 2 - 1
[[2,2],[1]]
 => [[2,1],[2]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 1 = 2 - 1
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2 = 3 - 1
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 3 = 4 - 1
[[5]]
 => [[5]]
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[[4,1]]
 => [[4],[1]]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 1 = 2 - 1
[[3,2]]
 => [[3],[2]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 1 = 2 - 1
[[3,1,1]]
 => [[3],[1],[1]]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2 = 3 - 1
[[2,2,1]]
 => [[2],[2],[1]]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2 = 3 - 1
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 3 = 4 - 1
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 4 = 5 - 1
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St001227
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [[1]]
 => [1]
 => [1,0,1,0]
 => 1
[[1],[1]]
 => [[1,1]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[2]]
 => [[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1]]
 => [[1],[1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[[1],[1],[1]]
 => [[1,1,1]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[2],[1]]
 => [[2,1]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,1],[1]]
 => [[1,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[3]]
 => [[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[2,1]]
 => [[2],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,1,1]]
 => [[1],[1],[1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[2],[1],[1]]
 => [[2,1,1]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[2],[2]]
 => [[2,2]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[[3],[1]]
 => [[3,1]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[2,1],[1]]
 => [[2,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[4]]
 => [[4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[3,1]]
 => [[3],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[2,2]]
 => [[2],[2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[[2,1,1]]
 => [[2],[1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[2],[2],[1]]
 => [[2,2,1]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[3],[1],[1]]
 => [[3,1,1]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[3],[2]]
 => [[3,2]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[2,1],[2]]
 => [[2,2],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4],[1]]
 => [[4,1]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[3,1],[1]]
 => [[3,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[2,2],[1]]
 => [[2,1],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[[5]]
 => [[5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[4,1]]
 => [[4],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[3,2]]
 => [[3],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[3,1,1]]
 => [[3],[1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[[2,2,1]]
 => [[2],[2],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[2],[2]]
 => [[2,2,2]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1],[1],[1],[1],[1]]
 => [[1,1,1,1,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2
[[1,1],[1,1],[1],[1]]
 => [[1,1,1,1],[1,1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[1,1],[1,1],[1,1]]
 => [[1,1,1],[1,1,1]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[3],[1],[1],[1]]
 => [[3,1,1,1]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[3],[2],[1]]
 => [[3,2,1]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[3],[3]]
 => [[3,3]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[4],[1],[1]]
 => [[4,1,1]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[4],[2]]
 => [[4,2]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[5],[1]]
 => [[5,1]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[6]]
 => [[6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,1,1,1,1]]
 => [[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[[1],[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[1],[1],[1],[1],[1]]
 => [[2,1,1,1,1,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[2],[1],[1],[1]]
 => [[2,2,1,1,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[2],[2],[1]]
 => [[2,2,2,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[3],[1],[1],[1],[1]]
 => [[3,1,1,1,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[3],[2],[1],[1]]
 => [[3,2,1,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[3],[2],[2]]
 => [[3,2,2]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[3],[3],[1]]
 => [[3,3,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[2,1],[1],[1],[1],[1]]
 => [[2,1,1,1,1],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[2,1],[2],[1],[1]]
 => [[2,2,1,1],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[2,1],[2],[2]]
 => [[2,2,2],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[4],[1],[1],[1]]
 => [[4,1,1,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[4],[2],[1]]
 => [[4,2,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[4],[3]]
 => [[4,3]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[3,1],[1],[1],[1]]
 => [[3,1,1,1],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[3,1],[2],[1]]
 => [[3,2,1],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[3,1],[3]]
 => [[3,3],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[5],[1],[1]]
 => [[5,1,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[5],[2]]
 => [[5,2]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[4,1],[1],[1]]
 => [[4,1,1],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[4,1],[2]]
 => [[4,2],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[6],[1]]
 => [[6,1]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[5,1],[1]]
 => [[5,1],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[1,1,1,1,1,1],[1]]
 => [[1,1],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 6
[[7]]
 => [[7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[6,1]]
 => [[6],[1]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 2
[[2,1,1,1,1,1]]
 => [[2],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 6
[[1,1,1,1,1,1,1]]
 => [[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7
[[1],[1],[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1,1,1]]
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[1],[1],[1],[1],[1],[1]]
 => [[2,1,1,1,1,1,1]]
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[2],[1],[1],[1],[1]]
 => [[2,2,1,1,1,1]]
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[2],[2],[1],[1]]
 => [[2,2,2,1,1]]
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[2],[2],[2],[2]]
 => [[2,2,2,2]]
 => [8]
 => [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],[1]]
 => [[1,1,1,1,1,1,1],[1]]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 2
[[1,1],[1,1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1],[1,1]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2
[[3],[1],[1],[1],[1],[1]]
 => [[3,1,1,1,1,1]]
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
[[3],[2],[1],[1],[1]]
 => [[3,2,1,1,1]]
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000329
Mapping
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 60%
Values
[[1]]
 => [[1]]
 => [1]
 => [1,0]
 => 0 = 1 - 1
[[1],[1]]
 => [[1,1]]
 => [2]
 => [1,0,1,0]
 => 0 = 1 - 1
[[2]]
 => [[2]]
 => [2]
 => [1,0,1,0]
 => 0 = 1 - 1
[[1,1]]
 => [[1],[1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[[1],[1],[1]]
 => [[1,1,1]]
 => [3]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[1]]
 => [[2,1]]
 => [3]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1,1],[1]]
 => [[1,1],[1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3]]
 => [[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[2,1]]
 => [[2],[1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1,1]]
 => [[1],[1],[1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[1],[1],[1]]
 => [[1,1,1,1]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[1],[1]]
 => [[2,1,1]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[2]]
 => [[2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[1,1],[1],[1]]
 => [[1,1,1],[1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1],[1,1]]
 => [[1,1],[1,1]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[[3],[1]]
 => [[3,1]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[2,1],[1]]
 => [[2,1],[1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1,1],[1]]
 => [[1,1],[1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[4]]
 => [[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[3,1]]
 => [[3],[1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,2]]
 => [[2],[2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,1,1]]
 => [[2],[1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1,1,1]]
 => [[1],[1],[1],[1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[1],[1],[1]]
 => [[2,1,1,1]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[2],[1]]
 => [[2,2,1]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[1,1],[1],[1],[1]]
 => [[1,1,1,1],[1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1],[1,1],[1]]
 => [[1,1,1],[1,1]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3],[1],[1]]
 => [[3,1,1]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[3],[2]]
 => [[3,2]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[2,1],[1],[1]]
 => [[2,1,1],[1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,1],[2]]
 => [[2,2],[1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,1],[1,1]]
 => [[2,1],[1,1]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[1,1,1],[1],[1]]
 => [[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1,1],[1,1]]
 => [[1,1],[1,1],[1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[4],[1]]
 => [[4,1]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[3,1],[1]]
 => [[3,1],[1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,2],[1]]
 => [[2,1],[2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,1,1],[1]]
 => [[2,1],[1],[1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1,1,1],[1]]
 => [[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[[5]]
 => [[5]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[4,1]]
 => [[4],[1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2]]
 => [[3],[2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,1,1]]
 => [[3],[1],[1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,1]]
 => [[2],[2],[1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[2,1,1,1]]
 => [[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[[1,1,1,1,1]]
 => [[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1]]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[1],[1],[1],[1]]
 => [[2,1,1,1,1]]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[2],[1],[1]]
 => [[2,2,1,1]]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[2],[1],[1],[1],[1],[1]]
 => [[2,1,1,1,1,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[2],[2],[1],[1],[1]]
 => [[2,2,1,1,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[2],[2],[2],[1]]
 => [[2,2,2,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[3],[1],[1],[1],[1]]
 => [[3,1,1,1,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[3],[2],[1],[1]]
 => [[3,2,1,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[3],[2],[2]]
 => [[3,2,2]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[3],[3],[1]]
 => [[3,3,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[2,1],[1],[1],[1],[1]]
 => [[2,1,1,1,1],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[2,1],[2],[1],[1]]
 => [[2,2,1,1],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[2,1],[2],[2]]
 => [[2,2,2],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[1,1,1],[1],[1],[1],[1]]
 => [[1,1,1,1,1],[1],[1]]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[[4],[1],[1],[1]]
 => [[4,1,1,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[4],[2],[1]]
 => [[4,2,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[4],[3]]
 => [[4,3]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[3,1],[1],[1],[1]]
 => [[3,1,1,1],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[3,1],[2],[1]]
 => [[3,2,1],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[3,1],[3]]
 => [[3,3],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[2,1,1],[1],[1],[1]]
 => [[2,1,1,1],[1],[1]]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[[2,1,1],[2],[1]]
 => [[2,2,1],[1],[1]]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[[1,1,1,1],[1],[1],[1]]
 => [[1,1,1,1],[1],[1],[1]]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 1
[[5],[1],[1]]
 => [[5,1,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[5],[2]]
 => [[5,2]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[4,1],[1],[1]]
 => [[4,1,1],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[4,1],[2]]
 => [[4,2],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[3,1,1],[1],[1]]
 => [[3,1,1],[1],[1]]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[[3,1,1],[2]]
 => [[3,2],[1],[1]]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[[2,1,1,1],[1],[1]]
 => [[2,1,1],[1],[1],[1]]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 1
[[2,1,1,1],[2]]
 => [[2,2],[1],[1],[1]]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 1
[[1,1,1,1,1],[1],[1]]
 => [[1,1,1],[1],[1],[1],[1]]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 1
[[6],[1]]
 => [[6,1]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[5,1],[1]]
 => [[5,1],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[4,1,1],[1]]
 => [[4,1],[1],[1]]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[[3,1,1,1],[1]]
 => [[3,1],[1],[1],[1]]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 1
[[2,1,1,1,1],[1]]
 => [[2,1],[1],[1],[1],[1]]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 1
[[1,1,1,1,1,1],[1]]
 => [[1,1],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 - 1
[[7]]
 => [[7]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[6,1]]
 => [[6],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[5,1,1]]
 => [[5],[1],[1]]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[[4,1,1,1]]
 => [[4],[1],[1],[1]]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 1
[[3,1,1,1,1]]
 => [[3],[1],[1],[1],[1]]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 1
[[2,1,1,1,1,1]]
 => [[2],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 - 1
[[1,1,1,1,1,1,1]]
 => [[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
 => [[1,1,1,1,1,1,1,1]]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[2],[1],[1],[1],[1],[1],[1]]
 => [[2,1,1,1,1,1,1]]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[2],[2],[1],[1],[1],[1]]
 => [[2,2,1,1,1,1]]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[2],[2],[2],[1],[1]]
 => [[2,2,2,1,1]]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[2],[2],[2],[2]]
 => [[2,2,2,2]]
 => [8]
 => [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]]
 => [[1,1,1,1,1,1,1],[1]]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.