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Your data matches 395 different statistics following compositions of up to 3 maps.
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Matching statistic: St001344
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
St001344: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The neighbouring number of a permutation.
For a permutation $\pi$, this is
$$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
Matching statistic: St000149
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1]
=> []
=> 0 = 1 - 1
[2,1] => [1] => [1]
=> []
=> 0 = 1 - 1
[1,2,3] => [1,2] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,3,2] => [1,2] => [1,1]
=> [1]
=> 0 = 1 - 1
[2,1,3] => [2,1] => [2]
=> []
=> 0 = 1 - 1
[2,3,1] => [2,1] => [2]
=> []
=> 0 = 1 - 1
[3,1,2] => [1,2] => [1,1]
=> [1]
=> 0 = 1 - 1
[3,2,1] => [2,1] => [2]
=> []
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,1,4] => [2,3,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,4,3,1] => [2,3,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[4,2,3,1] => [2,3,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,5,4,2] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> [1]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,5,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> 0 = 1 - 1
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000256
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1]
=> []
=> 0 = 1 - 1
[2,1] => [1] => [1]
=> []
=> 0 = 1 - 1
[1,2,3] => [1,2] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,3,2] => [1,2] => [1,1]
=> [1]
=> 0 = 1 - 1
[2,1,3] => [2,1] => [2]
=> []
=> 0 = 1 - 1
[2,3,1] => [2,1] => [2]
=> []
=> 0 = 1 - 1
[3,1,2] => [1,2] => [1,1]
=> [1]
=> 0 = 1 - 1
[3,2,1] => [2,1] => [2]
=> []
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,1,4] => [2,3,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,4,3,1] => [2,3,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[4,2,3,1] => [2,3,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,5,4,2] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> [1]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,5,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> 0 = 1 - 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000731
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,3,1] => [3,2,1] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[3,1,2] => [3,2,1] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[3,2,1] => [3,2,1] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,3,1,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[2,3,4,1] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[2,4,1,3] => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[2,4,3,1] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[3,1,4,2] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[3,2,4,1] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[3,4,1,2] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,1,3,2] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000732
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,3,1] => [3,2,1] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[3,1,2] => [3,2,1] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[3,2,1] => [3,2,1] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,3,1,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[2,3,4,1] => [4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[2,4,1,3] => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[2,4,3,1] => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[3,1,4,2] => [4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[3,2,4,1] => [4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[3,4,1,2] => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[4,1,3,2] => [4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[4,2,1,3] => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => 0 = 1 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => 0 = 1 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => 0 = 1 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => 0 = 1 - 1
Description
The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St001087
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[2,3,1] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[2,3,1,4] => [2,3,1] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[2,4,3,1] => [2,3,1] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[4,2,3,1] => [2,3,1] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 0 = 1 - 1
Description
The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.
Matching statistic: St001550
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,3,1] => [3,2,1] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[3,1,2] => [3,2,1] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[3,2,1] => [3,2,1] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,3,1,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[2,3,4,1] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[2,4,1,3] => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[2,4,3,1] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[3,1,4,2] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[3,2,4,1] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[3,4,1,2] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,1,3,2] => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St001714
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2]
=> 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [2]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [3]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [2,3,1] => [2,1]
=> 0 = 1 - 1
[2,1,3] => [1,3,2] => [2,3,1] => [2,1]
=> 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [3]
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [3]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => [3]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => [3,1]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [3,1]
=> 0 = 1 - 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => [2,2]
=> 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => [2,2]
=> 0 = 1 - 1
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => [2,2]
=> 0 = 1 - 1
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => [2,2]
=> 0 = 1 - 1
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => [2,2]
=> 0 = 1 - 1
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => [2,2]
=> 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [2,3,1,4] => [3,1]
=> 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => [3,1]
=> 0 = 1 - 1
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => [3,1]
=> 0 = 1 - 1
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => [2,2]
=> 0 = 1 - 1
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => [2,2]
=> 0 = 1 - 1
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => [3,1]
=> 0 = 1 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [2,3,1,4] => [3,1]
=> 0 = 1 - 1
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => [3,1]
=> 0 = 1 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [4,1]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [4,1]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [3,2]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => [4,1]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => [3,1,1]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => [3,2]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => [3,2]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,4,5,3] => [3,2]
=> 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => [3,2]
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2]
=> 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => [3,2]
=> 0 = 1 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [3,2]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,5,3,4] => [3,2]
=> 0 = 1 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,5,3,4] => [3,2]
=> 0 = 1 - 1
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St001162
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => ? ∊ {1,1}
[2,1] => [1] => [1] => [1] => ? ∊ {1,1}
[1,2,3] => [1,2] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,2,3] => [1,2,3] => 1
[1,3,4,2] => [1,3,2] => [1,2,3] => [1,2,3] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 1
[2,1,3,4] => [2,1,3] => [1,2,3] => [1,2,3] => 1
[2,1,4,3] => [2,1,3] => [1,2,3] => [1,2,3] => 1
[2,3,1,4] => [2,3,1] => [1,2,3] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,2,3] => [1,2,3] => 1
[2,4,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,3,2] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => [1,3,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => [1,3,2] => [3,1,2] => 1
[3,2,4,1] => [3,2,1] => [1,3,2] => [3,1,2] => 1
[3,4,1,2] => [3,1,2] => [1,3,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => [1,3,2] => [3,1,2] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 1
[4,2,1,3] => [2,1,3] => [1,2,3] => [1,2,3] => 1
[4,2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,3,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => [1,3,2] => [3,1,2] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 1
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,5,2,4,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
Description
The minimum jump of a permutation.
This is $\min_i |\pi_{i+1}-\pi_i|$, see [1].
Matching statistic: St000842
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => ? ∊ {1,1} + 1
[2,1] => [1] => [1] => [1] => ? ∊ {1,1} + 1
[1,2,3] => [1,2] => [1,2] => [1,2] => 2 = 1 + 1
[1,3,2] => [1,2] => [1,2] => [1,2] => 2 = 1 + 1
[2,1,3] => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[2,3,1] => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 2 = 1 + 1
[3,2,1] => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[1,4,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,2,4,1] => [3,2,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,4,1,2] => [3,1,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[4,1,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[1,5,2,4,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
Description
The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
The following 385 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001141The number of occurrences of hills of size 3 in a Dyck path. St001423The number of distinct cubes in a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001128The exponens consonantiae of a partition. St001282The number of graphs with the same chromatic polynomial. St001432The order dimension of the partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000068The number of minimal elements in a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000260The radius of a connected graph. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000517The Kreweras number of an integer partition. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001597The Frobenius rank of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000862The number of parts of the shifted shape of a permutation. St000878The number of ones minus the number of zeros of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000618The number of self-evacuating tableaux of given shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001924The number of cells in an integer partition whose arm and leg length coincide. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001568The smallest positive integer that does not appear twice in the partition. St001651The Frankl number of a lattice. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St000741The Colin de Verdière graph invariant. St001890The maximum magnitude of the Möbius function of a poset. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001593This is the number of standard Young tableaux of the given shifted shape. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000259The diameter of a connected graph. St000284The Plancherel distribution on integer partitions. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001518The number of graphs with the same ordinary spectrum as the given graph. St000287The number of connected components of a graph. St001765The number of connected components of the friends and strangers graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001330The hat guessing number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000456The monochromatic index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000759The smallest missing part in an integer partition. St000405The number of occurrences of the pattern 1324 in a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000297The number of leading ones in a binary word. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001768The number of reduced words of a signed permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000516The number of stretching pairs of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001513The number of nested exceedences of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001856The number of edges in the reduced word graph of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001555The order of a signed permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000889The number of alternating sign matrices with the same antidiagonal sums. St000478Another weight of a partition according to Alladi. St000181The number of connected components of the Hasse diagram for the poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St000069The number of maximal elements of a poset. St000629The defect of a binary word. St001429The number of negative entries in a signed permutation. St001621The number of atoms of a lattice. St001625The Möbius invariant of a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001889The size of the connectivity set of a signed permutation. St001616The number of neutral elements in a lattice. St000627The exponent of a binary word. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001863The number of weak excedances of a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000098The chromatic number of a graph. St001624The breadth of a lattice. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001395The number of strictly unfriendly partitions of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000636The hull number of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001029The size of the core of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001109The number of proper colourings of a graph with as few colours as possible. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001654The monophonic hull number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001430The number of positive entries in a signed permutation. St001829The common independence number of a graph. St000258The burning number of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001691The number of kings in a graph. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001927Sparre Andersen's number of positives of a signed permutation. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001410The minimal entry of a semistandard tableau. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001884The number of borders of a binary word. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000074The number of special entries. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000328The maximum number of child nodes in a tree. St000366The number of double descents of a permutation. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000876The number of factors in the Catalan decomposition of a binary word. St000894The trace of an alternating sign matrix. St000905The number of different multiplicities of parts of an integer composition. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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 $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001335The cardinality of a minimal cycle-isolating set of a graph. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001524The degree of symmetry of a binary word. St001557The number of inversions of the second entry of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001851The number of Hecke atoms of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000264The girth of a graph, which is not a tree. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000907The number of maximal antichains of minimal length in a poset. St000286The number of connected components of the complement of a graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001570The minimal number of edges to add to make a graph Hamiltonian.
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