Your data matches 86 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => 2 = 1 + 1
[[2,2]]
 => [2]
 => 1 = 0 + 1
[[1],[2]]
 => [1,1]
 => 2 = 1 + 1
[[1,3]]
 => [1,1]
 => 2 = 1 + 1
[[2,3]]
 => [1,1]
 => 2 = 1 + 1
[[3,3]]
 => [2]
 => 1 = 0 + 1
[[1],[3]]
 => [1,1]
 => 2 = 1 + 1
[[2],[3]]
 => [1,1]
 => 2 = 1 + 1
[[1,1,2]]
 => [2,1]
 => 2 = 1 + 1
[[1,2,2]]
 => [2,1]
 => 2 = 1 + 1
[[2,2,2]]
 => [3]
 => 1 = 0 + 1
[[1,1],[2]]
 => [2,1]
 => 2 = 1 + 1
[[1,2],[2]]
 => [2,1]
 => 2 = 1 + 1
[[1,4]]
 => [1,1]
 => 2 = 1 + 1
[[2,4]]
 => [1,1]
 => 2 = 1 + 1
[[3,4]]
 => [1,1]
 => 2 = 1 + 1
[[4,4]]
 => [2]
 => 1 = 0 + 1
[[1],[4]]
 => [1,1]
 => 2 = 1 + 1
[[2],[4]]
 => [1,1]
 => 2 = 1 + 1
[[3],[4]]
 => [1,1]
 => 2 = 1 + 1
[[1,1,3]]
 => [2,1]
 => 2 = 1 + 1
[[1,2,3]]
 => [1,1,1]
 => 3 = 2 + 1
[[1,3,3]]
 => [2,1]
 => 2 = 1 + 1
[[2,2,3]]
 => [2,1]
 => 2 = 1 + 1
[[2,3,3]]
 => [2,1]
 => 2 = 1 + 1
[[3,3,3]]
 => [3]
 => 1 = 0 + 1
[[1,1],[3]]
 => [2,1]
 => 2 = 1 + 1
[[1,2],[3]]
 => [1,1,1]
 => 3 = 2 + 1
[[1,3],[2]]
 => [1,1,1]
 => 3 = 2 + 1
[[1,3],[3]]
 => [2,1]
 => 2 = 1 + 1
[[2,2],[3]]
 => [2,1]
 => 2 = 1 + 1
[[2,3],[3]]
 => [2,1]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [1,1,1]
 => 3 = 2 + 1
[[1,1,1,2]]
 => [3,1]
 => 2 = 1 + 1
[[1,1,2,2]]
 => [2,2]
 => 2 = 1 + 1
[[1,2,2,2]]
 => [3,1]
 => 2 = 1 + 1
[[2,2,2,2]]
 => [4]
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [3,1]
 => 2 = 1 + 1
[[1,1,2],[2]]
 => [2,2]
 => 2 = 1 + 1
[[1,2,2],[2]]
 => [3,1]
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [2,2]
 => 2 = 1 + 1
[[1,5]]
 => [1,1]
 => 2 = 1 + 1
[[2,5]]
 => [1,1]
 => 2 = 1 + 1
[[3,5]]
 => [1,1]
 => 2 = 1 + 1
[[4,5]]
 => [1,1]
 => 2 = 1 + 1
[[5,5]]
 => [2]
 => 1 = 0 + 1
[[1],[5]]
 => [1,1]
 => 2 = 1 + 1
[[2],[5]]
 => [1,1]
 => 2 = 1 + 1
[[3],[5]]
 => [1,1]
 => 2 = 1 + 1
[[4],[5]]
 => [1,1]
 => 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ 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
[[1,2]]
 => [1,1]
 => [[1],[2]]
 => 1
[[2,2]]
 => [2]
 => [[1,2]]
 => 0
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => 1
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => 1
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => 1
[[3,3]]
 => [2]
 => [[1,2]]
 => 0
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => 1
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => 1
[[1,1,2]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[1,2,2]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => 0
[[1,1],[2]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[1,2],[2]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => 1
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => 1
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => 1
[[4,4]]
 => [2]
 => [[1,2]]
 => 0
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => 1
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => 1
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => 1
[[1,1,3]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2
[[1,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[2,2,3]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[2,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => 0
[[1,1],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2
[[1,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[2,2],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[2,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 1
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2
[[1,1,1,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1
[[1,2,2,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => 0
[[1,1,1],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1
[[1,2,2],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => 1
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => 1
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => 1
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => 1
[[5,5]]
 => [2]
 => [[1,2]]
 => 0
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => 1
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => 1
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => 1
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => 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: St000329
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[2,2]]
 => [2]
 => [1,0,1,0]
 => 0
[[1],[2]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[1,3]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[2,3]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[3,3]]
 => [2]
 => [1,0,1,0]
 => 0
[[1],[3]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[2],[3]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[1,1,2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[1,2,2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[2,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[[1,1],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[1,2],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[1,4]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[2,4]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[3,4]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[4,4]]
 => [2]
 => [1,0,1,0]
 => 0
[[1],[4]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[2],[4]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[3],[4]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[1,1,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[1,2,3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[1,3,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[2,2,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[2,3,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[3,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[[1,1],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[1,2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[1,3],[2]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[1,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[2,2],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[2,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[1],[2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[1,1,1,2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[[1,1,2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[[1,2,2,2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[[2,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[[1,1,1],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[[1,1,2],[2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[[1,2,2],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[[1,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[[1,5]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[2,5]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[3,5]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[4,5]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[5,5]]
 => [2]
 => [1,0,1,0]
 => 0
[[1],[5]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[2],[5]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[3],[5]]
 => [1,1]
 => [1,1,0,0]
 => 1
[[4],[5]]
 => [1,1]
 => [1,1,0,0]
 => 1
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ 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
[[1,2]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2,2]]
 => [2]
 => [1,1]
 => 1 = 0 + 1
[[1],[2]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[1,3]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2,3]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3,3]]
 => [2]
 => [1,1]
 => 1 = 0 + 1
[[1],[3]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2],[3]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[1,1,2]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[1,2,2]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[2,2,2]]
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[[1,1],[2]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[1,2],[2]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[1,4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2,4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3,4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[4,4]]
 => [2]
 => [1,1]
 => 1 = 0 + 1
[[1],[4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2],[4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3],[4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[1,1,3]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[1,2,3]]
 => [1,1,1]
 => [3]
 => 3 = 2 + 1
[[1,3,3]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[2,2,3]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[2,3,3]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[3,3,3]]
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[[1,1],[3]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[1,2],[3]]
 => [1,1,1]
 => [3]
 => 3 = 2 + 1
[[1,3],[2]]
 => [1,1,1]
 => [3]
 => 3 = 2 + 1
[[1,3],[3]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[2,2],[3]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[2,3],[3]]
 => [2,1]
 => [2,1]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [1,1,1]
 => [3]
 => 3 = 2 + 1
[[1,1,1,2]]
 => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[[1,1,2,2]]
 => [2,2]
 => [2,2]
 => 2 = 1 + 1
[[1,2,2,2]]
 => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[[2,2,2,2]]
 => [4]
 => [1,1,1,1]
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[[1,1,2],[2]]
 => [2,2]
 => [2,2]
 => 2 = 1 + 1
[[1,2,2],[2]]
 => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [2,2]
 => [2,2]
 => 2 = 1 + 1
[[1,5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2,5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3,5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[4,5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[5,5]]
 => [2]
 => [1,1]
 => 1 = 0 + 1
[[1],[5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2],[5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3],[5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[4],[5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
Description
The largest part of an integer partition.
Matching statistic: St000288
(load all 10 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ 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
[[1,2]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[2,2]]
 => [2]
 => 100 => 1 = 0 + 1
[[1],[2]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[1,3]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[2,3]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[3,3]]
 => [2]
 => 100 => 1 = 0 + 1
[[1],[3]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[2],[3]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[1,1,2]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1,2,2]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[2,2,2]]
 => [3]
 => 1000 => 1 = 0 + 1
[[1,1],[2]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1,2],[2]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1,4]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[2,4]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[3,4]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[4,4]]
 => [2]
 => 100 => 1 = 0 + 1
[[1],[4]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[2],[4]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[3],[4]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[1,1,3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1,2,3]]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
[[1,3,3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[2,2,3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[2,3,3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[3,3,3]]
 => [3]
 => 1000 => 1 = 0 + 1
[[1,1],[3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1,2],[3]]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
[[1,3],[2]]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
[[1,3],[3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[2,2],[3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[2,3],[3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1],[2],[3]]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
[[1,1,1,2]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[1,1,2,2]]
 => [2,2]
 => 1100 => 2 = 1 + 1
[[1,2,2,2]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[2,2,2,2]]
 => [4]
 => 10000 => 1 = 0 + 1
[[1,1,1],[2]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[1,1,2],[2]]
 => [2,2]
 => 1100 => 2 = 1 + 1
[[1,2,2],[2]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[1,1],[2,2]]
 => [2,2]
 => 1100 => 2 = 1 + 1
[[1,5]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[2,5]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[3,5]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[4,5]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[5,5]]
 => [2]
 => 100 => 1 = 0 + 1
[[1],[5]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[2],[5]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[3],[5]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[4],[5]]
 => [1,1]
 => 110 => 2 = 1 + 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
Mp00225: Semistandard tableaux —weight⟶ 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
[[1,2]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2,2]]
 => [2]
 => [1,1]
 => 1 = 0 + 1
[[1],[2]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[1,3]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2,3]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3,3]]
 => [2]
 => [1,1]
 => 1 = 0 + 1
[[1],[3]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2],[3]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[1,1,2]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[1,2,2]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[2,2,2]]
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[[1,1],[2]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[1,2],[2]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[1,4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2,4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3,4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[4,4]]
 => [2]
 => [1,1]
 => 1 = 0 + 1
[[1],[4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2],[4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3],[4]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[1,1,3]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[1,2,3]]
 => [1,1,1]
 => [2,1]
 => 3 = 2 + 1
[[1,3,3]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[2,2,3]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[2,3,3]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[3,3,3]]
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[[1,1],[3]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[1,2],[3]]
 => [1,1,1]
 => [2,1]
 => 3 = 2 + 1
[[1,3],[2]]
 => [1,1,1]
 => [2,1]
 => 3 = 2 + 1
[[1,3],[3]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[2,2],[3]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[2,3],[3]]
 => [2,1]
 => [3]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [1,1,1]
 => [2,1]
 => 3 = 2 + 1
[[1,1,1,2]]
 => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[[1,1,2,2]]
 => [2,2]
 => [4]
 => 2 = 1 + 1
[[1,2,2,2]]
 => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[[2,2,2,2]]
 => [4]
 => [1,1,1,1]
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[[1,1,2],[2]]
 => [2,2]
 => [4]
 => 2 = 1 + 1
[[1,2,2],[2]]
 => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [2,2]
 => [4]
 => 2 = 1 + 1
[[1,5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2,5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3,5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[4,5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[5,5]]
 => [2]
 => [1,1]
 => 1 = 0 + 1
[[1],[5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[2],[5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[3],[5]]
 => [1,1]
 => [2]
 => 2 = 1 + 1
[[4],[5]]
 => [1,1]
 => [2]
 => 2 = 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
(load all 3 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ 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
[[1,2]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[2,2]]
 => [2]
 => [[1,2]]
 => 1 = 0 + 1
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[3,3]]
 => [2]
 => [[1,2]]
 => 1 = 0 + 1
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[1,1,2]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2,2]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[[1,1],[2]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2],[2]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[4,4]]
 => [2]
 => [[1,2]]
 => 1 = 0 + 1
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[1,1,3]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[[1,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[2,2,3]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[2,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[[1,1],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[[1,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[2,2],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[2,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[[1,1,1,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2 = 1 + 1
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2 = 1 + 1
[[1,2,2,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2 = 1 + 1
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2 = 1 + 1
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2 = 1 + 1
[[1,2,2],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2 = 1 + 1
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[5,5]]
 => [2]
 => [[1,2]]
 => 1 = 0 + 1
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2,2]]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3,3]]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[1,1,2]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[1,2,2]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[[1,1],[2]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[1,2],[2]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[4,4]]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[1,1,3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[[1,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[2,2,3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[2,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[[1,1],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[[1,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[2,2],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[2,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[[1,1,1,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[[1,2,2,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
[[1,1,1],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[[1,2,2],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[5,5]]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000053
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[2,2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[1],[2]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[1,3]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[2,3]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,3]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[1],[3]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[2],[3]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[1,1,2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1,2,2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[1,1],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1,2],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1,4]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[2,4]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,4]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,4]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[1],[4]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[2],[4]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3],[4]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[1,1,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1,2,3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[[1,3,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2,2,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2,3,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[3,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[1,1],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1,2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[[1,3],[2]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[[1,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2,2],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[[1,1,1,2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,1,2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[[1,2,2,2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[2,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[1,1,1],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,1,2],[2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[[1,2,2],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[[1,5]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[2,5]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,5]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,5]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[5,5]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[1],[5]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[2],[5]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3],[5]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4],[5]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000155
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[2,2]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0
[[1],[2]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[1,3]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[2,3]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[3,3]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0
[[1],[3]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[2],[3]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[1,1,2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[1,2,2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[2,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[1,1],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[1,2],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[1,4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[2,4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[3,4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[4,4]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0
[[1],[4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[2],[4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[3],[4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[1,1,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[1,2,3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[[1,3,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[2,2,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[2,3,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[3,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[1,1],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[1,2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[[1,3],[2]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[[1,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[2,2],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[2,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[1],[2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[[1,1,1,2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[[1,1,2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[[1,2,2,2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[[2,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[[1,1,1],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[[1,1,2],[2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[[1,2,2],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[[1,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[[1,5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[2,5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[3,5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[4,5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[5,5]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0
[[1],[5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[2],[5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[3],[5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[4],[5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
The following 76 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000216The absolute length of a permutation. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001777The number of weak descents in an integer composition. St000007The number of saliances of the permutation. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001530The depth of a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001152The number of pairs with even minimum in a perfect matching. St000668The least common multiple of the parts of the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001401The number of distinct entries in a semistandard tableau. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000259The diameter of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001877Number of indecomposable injective modules with projective dimension 2. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000264The girth of a graph, which is not a tree. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.