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Your data matches 91 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000147
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ 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%
Mp00204: Permutations —LLPS⟶ 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
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [2]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [3]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [3,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [3,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [3,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> [3,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> [3,2]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [3,2]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [3,2]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> [3,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [3,2]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> [3,2]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> [4,1]
=> 4
Description
The largest part of an integer partition.
Matching statistic: St000378
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ 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%
Mp00204: Permutations —LLPS⟶ 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
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [2]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [2,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [3]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [3]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [3]
=> 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [3]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [3,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [2,2]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [2,2]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [2,2]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> [2,2]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [2,2]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [4]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [2,2]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [2,2]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [4]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> [2,2]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [3,2]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> [2,2,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> [3,1,1]
=> 4
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: St001250
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 4
[]
=> [] => []
=> ? = 0
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St000876
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> 1 => 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 11 => 2
[1,1,0,0]
=> [2,1] => [2]
=> 0 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 111 => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 01 => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 01 => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 01 => 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 01 => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 011 => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 011 => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 011 => 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 011 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 011 => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 00 => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 011 => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 011 => 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 011 => 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 011 => 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 00 => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 011 => 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 011 => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 0111 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 0111 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 0111 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 001 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 001 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 0111 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 0111 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 001 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 001 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 001 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 001 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 001 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 001 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 0111 => 4
[]
=> [] => []
=> ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => ? = 11
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,9,10,11,12,2] => [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
=> [11,1,2,3,4,5,6,7,8,9,12,10] => [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,1,2,3,4,5,6,7,8,9,12,11] => [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? = 10
[1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [7,1,8,9,10,11,12,2,3,4,5,6] => [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? = 10
Description
The number of factors in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000885
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000885: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000885: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> 1 => 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 11 => 2
[1,1,0,0]
=> [2,1] => [2]
=> 0 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 111 => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 01 => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 01 => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 01 => 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 01 => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 011 => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 011 => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 011 => 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 011 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 011 => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 00 => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 011 => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 011 => 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 011 => 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 011 => 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 00 => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 011 => 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 011 => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 0111 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 0111 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 0111 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 001 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 001 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 0111 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 0111 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 0111 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 001 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 001 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 001 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 001 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 001 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 001 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 0111 => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 0111 => 4
[]
=> [] => []
=> ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => ? = 11
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,9,10,11,12,2] => [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
=> [11,1,2,3,4,5,6,7,8,9,12,10] => [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,1,2,3,4,5,6,7,8,9,12,11] => [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? = 10
[1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [7,1,8,9,10,11,12,2,3,4,5,6] => [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? = 10
Description
The number of critical steps in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of critical steps $\ell + m$ in the Catalan factorisation.
The distribution of this statistic on words of length $n$ is
$$
(n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}.
$$
Matching statistic: St000288
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1]
=> 10 => 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 110 => 2
[1,1,0,0]
=> [2,1] => [2]
=> 100 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1110 => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1010 => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1010 => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1010 => 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1010 => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 11110 => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 10110 => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 10110 => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 10110 => 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 10110 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 10110 => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1100 => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 10110 => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 10110 => 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 10110 => 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 10110 => 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 1100 => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 10110 => 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 10110 => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 111110 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 101110 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 101110 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 101110 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 101110 => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 101110 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 11010 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 101110 => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 101110 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 101110 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 101110 => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 11010 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 101110 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 101110 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 101110 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 11010 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 11010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 11010 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 11010 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 101110 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 11010 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 101110 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 101110 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 101110 => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 101110 => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 11010 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 101110 => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 101110 => 4
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[]
=> [] => []
=> => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,8,10,11,9] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,7,8,11,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,12,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,7,8,9,12,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,9,10,11,12,2] => [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 10
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,6,7,8,9,11,1,2,3,4,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [9,11,1,2,3,4,5,6,7,8,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [10,11,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,7,8,10,11,1,9] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,8,9,10,11,1,3] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,8,9,10,11,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,11,1,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
=> [11,1,2,3,4,5,6,7,8,9,12,10] => [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 10
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,12,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,6,7,8,9,10,1,2,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,9,10,1,2,3,4,5,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,10,11,2,3,4,5,6] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,11,1,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,8,9,10,11,12,1,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,8,9,10,12,1,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [8,11,1,2,3,4,5,6,7,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9,10,11,12] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,1,2,3,4,5,6,7,8,9,12,11] => [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11,12,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,8,9,10,11,1,3,4,5,6,7] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,6,7,8,9,11,2,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [6,7,8,9,10,12,1,2,3,4,5,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 11
[1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [7,1,8,9,10,11,12,2,3,4,5,6] => [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 10
[1,1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [5,11,1,2,3,4,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 10
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,4,7,6,9,8,11,12,10] => [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000733
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1]
=> [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [[1],[2]]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> [[1,2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[]
=> [] => []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,8,10,11,9] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,7,8,11,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,12,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,7,8,9,12,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,8,9,10,11,2] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,9,10,11,12,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,6,7,8,9,11,1,2,3,4,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,1,7,8,9,10,11,2,3,4,5] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [9,11,1,2,3,4,5,6,7,8,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1,11,2,3,4,5,6,7,8,9] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [10,11,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,7,8,10,11,1,9] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,8,9,10,11,1,3] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,8,9,10,11,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,11,1,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
=> [11,1,2,3,4,5,6,7,8,9,12,10] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,12,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,8,1,9,10,11,7] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0]
=> [10,1,2,11,3,4,5,6,7,8,9] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8,11,10] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 6
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,6,7,8,9,10,1,2,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,7,8,9,10,1,11,2] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,1,7,8,9,10,11,5] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,9,10,1,2,3,4,5,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,10,11,2,3,4,5,6] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,11,1,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,8,9,10,11,12,1,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,8,9,10,12,1,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [8,11,1,2,3,4,5,6,7,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,3,4,6,8,5,7,10,9,11] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,2,3,4,7,5,8,6,10,9,11] => [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 8
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9,10,11,12] => [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,1,2,3,4,5,6,7,8,9,12,11] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11,12,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000507
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [[1,2]]
=> 2
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [3,1,2] => [[1,3],[2]]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 7
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 6
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 6
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ? = 6
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ? = 6
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ? = 6
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ? = 6
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ? = 6
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ? = 6
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => ?
=> ? = 6
[1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,5,7,2,3,4,6,8] => ?
=> ? = 7
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ? = 6
[1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5,8] => ?
=> ? = 6
[1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,1,6,7,3,8,4,5] => ?
=> ? = 5
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4,8] => ?
=> ? = 6
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,3,6,7,1,4,5,8] => ?
=> ? = 7
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,6,7,1,4,8,5] => ?
=> ? = 6
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,6,8,1,4,5,7] => ?
=> ? = 7
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,4,1,6,7,3,5,8] => ?
=> ? = 6
[1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,7,3,8] => ?
=> ? = 6
[1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,4,5,7,1,3,8,6] => ?
=> ? = 6
[1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,6,1,3,7,8,5] => ?
=> ? = 6
[1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [2,4,6,1,7,8,3,5] => ?
=> ? = 6
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [2,4,6,7,1,8,3,5] => ?
=> ? = 6
[1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [2,4,7,1,3,8,5,6] => ?
=> ? = 6
[1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,8,4,7] => ?
=> ? = 5
[1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,8,4,6] => ?
=> ? = 5
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [2,5,7,1,3,8,4,6] => ?
=> ? = 6
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [3,1,2,5,6,4,7,8] => ?
=> ? = 6
[1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,1,2,5,7,8,4,6] => ?
=> ? = 6
[1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,4,7,2,8,5,6] => ?
=> ? = 5
[1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,5,2,4,6,7,8] => ?
=> ? = 6
[1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,5,2,4,6,8,7] => ?
=> ? = 5
[1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,5,2,4,7,8,6] => ?
=> ? = 5
[1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,4,1,7,2,5,8,6] => ?
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> [3,4,5,1,7,2,6,8] => ?
=> ? = 6
[1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> [3,5,7,1,2,8,4,6] => ?
=> ? = 6
[1,1,1,1,0,0,1,0,1,1,0,0,0,0,1,0]
=> [4,1,5,7,2,3,6,8] => ?
=> ? = 6
[1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
=> [4,5,1,7,2,3,6,8] => ?
=> ? = 6
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [4,5,7,1,2,3,8,6] => ?
=> ? = 6
[1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [4,6,1,7,2,3,8,5] => ?
=> ? = 5
[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [5,6,1,2,7,3,4,8] => ?
=> ? = 6
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,8,3,4] => ?
=> ? = 5
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [[1,2,3,6,7,8,9],[4,5]]
=> ? = 8
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 10
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000394
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,5,8,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3,5,4,7,8,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,3,5,6,4,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,4,8,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,6,4,5,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,4,7,8,5] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,4,8,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,4,5,8,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5,7,6,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,3,5,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,8,7] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,2,4,3,6,7,5,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,3,6,8,5,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,7,5,6,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,2,4,3,7,8,5,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,5,6,7] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,4,5,3,7,8,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,6,3,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,3,8,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,2,4,6,3,5,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,2,4,7,3,5,8,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,5,3,4,7,6,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,3,4,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,2,5,3,6,4,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,2,5,3,6,4,8,7] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,2,5,3,6,7,4,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,2,5,3,7,4,6,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,2,5,3,7,4,8,6] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,8,4,6,7] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,3,7,8,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,2,5,6,7,3,8,4] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,3,4,8,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,3,4,7,5,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,6,3,4,8,5,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,6,3,7,4,5,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,6,3,7,4,8,5] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,2,6,3,7,8,4,5] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,6,3,8,4,5,7] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,7,3,4,8,5,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,7,3,8,4,5,6] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,5,7,6,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5,8,7] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,5,8,6] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
The following 81 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000676The number of odd rises of a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000245The number of ascents of a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000925The number of topologically connected components of a set partition. St001286The annihilation number of a graph. St000482The (zero)-forcing number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000672The number of minimal elements in Bruhat order not less than the permutation. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000167The number of leaves of an ordered tree. St000996The number of exclusive left-to-right maxima of a permutation. St000308The height of the tree associated to a permutation. St000007The number of saliances of the permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000703The number of deficiencies of a permutation. St000553The number of blocks of a graph. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St000021The number of descents of a permutation. St000015The number of peaks of a Dyck path. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000080The rank of the poset. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001480The number of simple summands of the module J^2/J^3. St001427The number of descents of a signed permutation. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001863The number of weak excedances of a signed permutation. St000871The number of very big ascents of a permutation. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001946The number of descents in a parking function.
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