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Your data matches 70 different statistics following compositions of up to 3 maps.
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Matching statistic: St000459
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Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,1]
=> 2
[[2,2]]
=> [2]
=> 2
[[1],[2]]
=> [1,1]
=> 2
[[1,3]]
=> [1,1]
=> 2
[[2,3]]
=> [1,1]
=> 2
[[3,3]]
=> [2]
=> 2
[[1],[3]]
=> [1,1]
=> 2
[[2],[3]]
=> [1,1]
=> 2
[[1,1,2]]
=> [2,1]
=> 3
[[1,2,2]]
=> [2,1]
=> 3
[[2,2,2]]
=> [3]
=> 3
[[1,1],[2]]
=> [2,1]
=> 3
[[1,2],[2]]
=> [2,1]
=> 3
[[1,4]]
=> [1,1]
=> 2
[[2,4]]
=> [1,1]
=> 2
[[3,4]]
=> [1,1]
=> 2
[[4,4]]
=> [2]
=> 2
[[1],[4]]
=> [1,1]
=> 2
[[2],[4]]
=> [1,1]
=> 2
[[3],[4]]
=> [1,1]
=> 2
[[1,1,3]]
=> [2,1]
=> 3
[[1,2,3]]
=> [1,1,1]
=> 3
[[1,3,3]]
=> [2,1]
=> 3
[[2,2,3]]
=> [2,1]
=> 3
[[2,3,3]]
=> [2,1]
=> 3
[[3,3,3]]
=> [3]
=> 3
[[1,1],[3]]
=> [2,1]
=> 3
[[1,2],[3]]
=> [1,1,1]
=> 3
[[1,3],[2]]
=> [1,1,1]
=> 3
[[1,3],[3]]
=> [2,1]
=> 3
[[2,2],[3]]
=> [2,1]
=> 3
[[2,3],[3]]
=> [2,1]
=> 3
[[1],[2],[3]]
=> [1,1,1]
=> 3
[[1,1,1,2]]
=> [3,1]
=> 4
[[1,1,2,2]]
=> [2,2]
=> 3
[[1,2,2,2]]
=> [3,1]
=> 4
[[2,2,2,2]]
=> [4]
=> 4
[[1,1,1],[2]]
=> [3,1]
=> 4
[[1,1,2],[2]]
=> [2,2]
=> 3
[[1,2,2],[2]]
=> [3,1]
=> 4
[[1,1],[2,2]]
=> [2,2]
=> 3
[[1,5]]
=> [1,1]
=> 2
[[2,5]]
=> [1,1]
=> 2
[[3,5]]
=> [1,1]
=> 2
[[4,5]]
=> [1,1]
=> 2
[[5,5]]
=> [2]
=> 2
[[1],[5]]
=> [1,1]
=> 2
[[2],[5]]
=> [1,1]
=> 2
[[3],[5]]
=> [1,1]
=> 2
[[4],[5]]
=> [1,1]
=> 2
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000395
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: 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]
=> [1,1,0,0]
=> 2
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2,3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,3],[2]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,1,1,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[1,1,2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,2,2,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[1,1,2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001004
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001004: 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]
=> [1,2] => 2
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[1,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[2,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[1,2,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[1,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[2,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[1,2,3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 3
[[1,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[2,2,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[2,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[1,2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 3
[[1,3],[2]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 3
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 3
[[1,1,1,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 4
[[1,1,2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[[1,2,2,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 4
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 4
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 4
[[1,1,2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 4
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[[1,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[2,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[4,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St001348
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001348: 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]
=> [1,1,0,0]
=> 2
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,2,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,2,3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[[1,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[2,2,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[2,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[[1,3],[2]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[[1,1,1,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[1,1,2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,2,2,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[1,1,2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
Description
The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St000806
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000806: Integer compositions ⟶ 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] => 3 = 2 + 1
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [1,1] => 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[1,3]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[2,3]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [1,1] => 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[1,2,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[1,4]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[2,4]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[3,4]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [1,1] => 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[1,2,3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[2,2,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[2,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[1,2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 5 = 4 + 1
[[1,1,2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3] => 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 5 = 4 + 1
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 5 = 4 + 1
[[1,1,2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3] => 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 5 = 4 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3] => 4 = 3 + 1
[[1,5]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[2,5]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[3,5]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[4,5]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [1,1] => 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St001267
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[2,2]]
=> [2]
=> [[1,2]]
=> 0 => 1 = 2 - 1
[[1],[2]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[1,3]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[2,3]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[3,3]]
=> [2]
=> [[1,2]]
=> 0 => 1 = 2 - 1
[[1],[3]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[2],[3]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[1,1,2]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[1,2,2]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[2,2,2]]
=> [3]
=> [[1,2,3]]
=> 00 => 2 = 3 - 1
[[1,1],[2]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[1,2],[2]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[1,4]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[2,4]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[3,4]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[4,4]]
=> [2]
=> [[1,2]]
=> 0 => 1 = 2 - 1
[[1],[4]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[2],[4]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[3],[4]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[1,1,3]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[1,2,3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 2 = 3 - 1
[[1,3,3]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[2,2,3]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[2,3,3]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[3,3,3]]
=> [3]
=> [[1,2,3]]
=> 00 => 2 = 3 - 1
[[1,1],[3]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[1,2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 2 = 3 - 1
[[1,3],[2]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 2 = 3 - 1
[[1,3],[3]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[2,2],[3]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[2,3],[3]]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 2 = 3 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 2 = 3 - 1
[[1,1,1,2]]
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 3 = 4 - 1
[[1,1,2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 2 = 3 - 1
[[1,2,2,2]]
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 3 = 4 - 1
[[2,2,2,2]]
=> [4]
=> [[1,2,3,4]]
=> 000 => 3 = 4 - 1
[[1,1,1],[2]]
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 3 = 4 - 1
[[1,1,2],[2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 2 = 3 - 1
[[1,2,2],[2]]
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 3 = 4 - 1
[[1,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 2 = 3 - 1
[[1,5]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[2,5]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[3,5]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[4,5]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[5,5]]
=> [2]
=> [[1,2]]
=> 0 => 1 = 2 - 1
[[1],[5]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[2],[5]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[3],[5]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[4],[5]]
=> [1,1]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
Description
The length of the Lyndon factorization of the binary word.
The Lyndon factorization of a finite word w is its unique factorization as a non-increasing product of Lyndon words, i.e., $w = l_1\dots l_n$ where each $l_i$ is a Lyndon word and $l_1 \geq\dots\geq l_n$.
Matching statistic: St001020
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[[1,2]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2,3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,3],[2]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,1,1,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[1,1,2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,2,2,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[1,1,2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,1,1,1,1,1,1,2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {8,8,8,8,8}
[[1,2,2,2,2,2,2,2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {8,8,8,8,8}
[[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {8,8,8,8,8}
[[1,1,1,1,1,1,1],[2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {8,8,8,8,8}
[[1,2,2,2,2,2,2],[2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {8,8,8,8,8}
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000998
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[[1,2]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[1,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[2,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[2,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[3,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2,3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,2,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[1,1,2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[1,1,2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[2,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[3,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[4,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[1,1,1,1,1,1,1,2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {8,8,8,8,8} + 1
[[1,2,2,2,2,2,2,2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {8,8,8,8,8} + 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,8,8,8,8} + 1
[[1,1,1,1,1,1,1],[2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {8,8,8,8,8} + 1
[[1,2,2,2,2,2,2],[2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {8,8,8,8,8} + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000144
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 98%●distinct values known / distinct values provided: 71%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 98%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2,3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,3],[2]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,1,1,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[1,1,2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,2,2,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[1,1,2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[1,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4,5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[1,1,1,1,1,1,2]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7}
[[1,2,2,2,2,2,2]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7}
[[2,2,2,2,2,2,2]]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,7,7,7,7}
[[1,1,1,1,1,1],[2]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7}
[[1,2,2,2,2,2],[2]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7}
[[1,1,1,1,1,1,3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,2,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2,2,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,3,3,3,3,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,3,3,3,3,3,3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[2,2,2,2,2,2,3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[2,3,3,3,3,3,3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[3,3,3,3,3,3,3]]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,1],[3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,3],[2]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2,2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2,3],[2]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,3,3,3,3],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,3,3,3,3,3],[2]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,3,3,3,3,3],[3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[2,2,2,2,2,2],[3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[2,3,3,3,3,3],[3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1],[2,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2],[2,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,3,3,3],[3,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,3,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,1,1,2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,1,1,1,1,2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,2,2,2,2,2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,2,2,2,2,2,2,2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,1,1,1,1,1],[2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,1,1,1,1,2],[2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,2,2,2,2,2],[2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,2,2,2,2,2,2],[2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000288
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 98%●distinct values known / distinct values provided: 71%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 98%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[2,2]]
=> [2]
=> [1,0,1,0]
=> 1010 => 2
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[1,3]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[2,3]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[3,3]]
=> [2]
=> [1,0,1,0]
=> 1010 => 2
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[1,2,2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 3
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[1,4]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[2,4]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[3,4]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[4,4]]
=> [2]
=> [1,0,1,0]
=> 1010 => 2
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[1,2,3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 3
[[1,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[2,2,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[2,3,3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 3
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[1,2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 3
[[1,3],[2]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 3
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 3
[[1,1,1,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 4
[[1,1,2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 3
[[1,2,2,2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 4
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 4
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 4
[[1,1,2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 3
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 4
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 3
[[1,5]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[2,5]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[3,5]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[4,5]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[5,5]]
=> [2]
=> [1,0,1,0]
=> 1010 => 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 2
[[1,1,1,1,1,1,2]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7}
[[1,2,2,2,2,2,2]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7}
[[2,2,2,2,2,2,2]]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? ∊ {7,7,7,7,7}
[[1,1,1,1,1,1],[2]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7}
[[1,2,2,2,2,2],[2]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7}
[[1,1,1,1,1,1,3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,2,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2,2,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,3,3,3,3,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,3,3,3,3,3,3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[2,2,2,2,2,2,3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[2,3,3,3,3,3,3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[3,3,3,3,3,3,3]]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,1],[3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,3],[2]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2,2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2,3],[2]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,3,3,3,3],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,3,3,3,3,3],[2]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,3,3,3,3,3],[3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[2,2,2,2,2,2],[3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[2,3,3,3,3,3],[3]]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1],[2,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2],[2,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,3,3,3],[3,3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,2,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,3,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[[1,1,1,1,1,1,1,2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,1,1,1,1,2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,2,2,2,2,2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,2,2,2,2,2,2,2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[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]
=> 1010101010101010 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,1,1,1,1,1],[2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,1,1,1,1,2],[2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,2,2,2,2,2],[2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,2,2,2,2,2,2],[2]]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? ∊ {7,7,7,7,7,7,8,8,8,8,8}
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000924The number of topologically connected components of a perfect matching. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000026The position of the first return of a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000019The cardinality of the support of a permutation. St000863The length of the first row of the shifted shape of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001480The number of simple summands of the module J^2/J^3. St001958The degree of the polynomial interpolating the values of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001468The smallest fixpoint of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001621The number of atoms of a lattice. St000550The number of modular elements of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000527The width of the poset. St000845The maximal number of elements covered by an element in a poset. St000632The jump number of the poset. St001875The number of simple modules with projective dimension at most 1. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001926Sparre Andersen's position of the maximum of a signed permutation. St000168The number of internal nodes of an ordered tree. St000782The indicator function of whether a given perfect matching is an L & P matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000422The energy of a graph, if it is integral. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau. St000101The cocharge of a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St000454The largest eigenvalue of a graph if it is integral. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000102The charge of a semistandard tableau. St000327The number of cover relations in a poset. St000635The number of strictly order preserving maps of a poset into itself. St001964The interval resolution global dimension of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph.
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