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Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000742
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000742: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000742: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,3,2] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,4,2] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,1,4,3,2] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6,3,5,1,4,2] => 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,6,5,1,4,3] => 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,5,2] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [6,4,1,5,3,2] => 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,6,4,2] => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,5,4,2] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 2
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 2
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 2
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 2
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 2
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 2
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 2
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 2
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 2
Description
The number of big ascents of a permutation after prepending zero.
Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$.
It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]).
G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]].
[[St000646]] is the statistic without the border condition $\pi(0) = 0$.
Matching statistic: St000667
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ? = 1 - 1
[2]
=> []
=> ?
=> ? = 3 - 1
[1,1]
=> [1]
=> []
=> ? = 2 - 1
[3]
=> []
=> ?
=> ? = 3 - 1
[2,1]
=> [1]
=> []
=> ? = 2 - 1
[1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4]
=> []
=> ?
=> ? = 3 - 1
[3,1]
=> [1]
=> []
=> ? = 2 - 1
[2,2]
=> [2]
=> []
=> ? = 3 - 1
[2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,1]
=> [1]
=> []
=> ? = 3 - 1
[3,2]
=> [2]
=> []
=> ? = 3 - 1
[3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,2]
=> [2]
=> []
=> ? = 3 - 1
[4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,3]
=> [3]
=> []
=> ? = 3 - 1
[3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,3]
=> [3]
=> []
=> ? = 3 - 1
[4,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[3,3,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[3,2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,3,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[4,2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[3,3,2]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
[4,3,2]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> 1 = 2 - 1
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> 1 = 2 - 1
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> 1 = 2 - 1
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> 1 = 2 - 1
[5,5,2,1,1,1]
=> [5,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> 1 = 2 - 1
[5,5,3,2,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> 1 = 2 - 1
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> 1 = 2 - 1
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> 1 = 2 - 1
[]
=> ?
=> ?
=> ? = 0 - 1
[5,5,4,3,2,1]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> 1 = 2 - 1
[5,5,3,2,2,1]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> 1 = 2 - 1
[5,5,4,2,1,1]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> 1 = 2 - 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000439
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? = 3
[1,1]
=> [1]
=> []
=> []
=> ? = 2
[3]
=> []
=> ?
=> ?
=> ? = 3
[2,1]
=> [1]
=> []
=> []
=> ? = 2
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 2
[4]
=> []
=> ?
=> ?
=> ? = 3
[3,1]
=> [1]
=> []
=> []
=> ? = 2
[2,2]
=> [2]
=> []
=> []
=> ? = 3
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[4,1]
=> [1]
=> []
=> []
=> ? = 3
[3,2]
=> [2]
=> []
=> []
=> ? = 3
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[4,2]
=> [2]
=> []
=> []
=> ? = 3
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 2
[3,3]
=> [3]
=> []
=> []
=> ? = 3
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[4,3]
=> [3]
=> []
=> []
=> ? = 3
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[5,5,2,1,1,1]
=> [5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[5,5,3,2,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[]
=> ?
=> ?
=> ?
=> ? = 0
[5,5,4,3,2,1]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
[5,5,3,2,2,1]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[5,5,4,2,1,1]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000026
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1 - 1
[2]
=> []
=> ?
=> ?
=> ? = 3 - 1
[1,1]
=> [1]
=> []
=> []
=> ? = 2 - 1
[3]
=> []
=> ?
=> ?
=> ? = 3 - 1
[2,1]
=> [1]
=> []
=> []
=> ? = 2 - 1
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[4]
=> []
=> ?
=> ?
=> ? = 3 - 1
[3,1]
=> [1]
=> []
=> []
=> ? = 2 - 1
[2,2]
=> [2]
=> []
=> []
=> ? = 3 - 1
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1]
=> [1]
=> []
=> []
=> ? = 3 - 1
[3,2]
=> [2]
=> []
=> []
=> ? = 3 - 1
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[4,2]
=> [2]
=> []
=> []
=> ? = 3 - 1
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,3]
=> [3]
=> []
=> []
=> ? = 3 - 1
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3]
=> [3]
=> []
=> []
=> ? = 3 - 1
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[5,5,2,1,1,1]
=> [5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[5,5,3,2,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[]
=> ?
=> ?
=> ?
=> ? = 0 - 1
[5,5,4,3,2,1]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[5,5,3,2,2,1]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[5,5,4,2,1,1]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000791
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1 - 2
[2]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[3]
=> []
=> ?
=> ?
=> ? = 3 - 2
[2,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[4]
=> []
=> ?
=> ?
=> ? = 3 - 2
[3,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[2,2]
=> [2]
=> []
=> []
=> ? = 3 - 2
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[4,1]
=> [1]
=> []
=> []
=> ? = 3 - 2
[3,2]
=> [2]
=> []
=> []
=> ? = 3 - 2
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[4,2]
=> [2]
=> []
=> []
=> ? = 3 - 2
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[3,3]
=> [3]
=> []
=> []
=> ? = 3 - 2
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[4,3]
=> [3]
=> []
=> []
=> ? = 3 - 2
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[5,5,2,1,1,1]
=> [5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[5,5,3,2,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[]
=> ?
=> ?
=> ?
=> ? = 0 - 2
[5,5,4,3,2,1]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[5,5,3,2,2,1]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[5,5,4,2,1,1]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
Description
The number of pairs of left tunnels, one strictly containing the other, of a Dyck path.
The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St001217
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1 - 2
[2]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[3]
=> []
=> ?
=> ?
=> ? = 3 - 2
[2,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[4]
=> []
=> ?
=> ?
=> ? = 3 - 2
[3,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[2,2]
=> [2]
=> []
=> []
=> ? = 3 - 2
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[4,1]
=> [1]
=> []
=> []
=> ? = 3 - 2
[3,2]
=> [2]
=> []
=> []
=> ? = 3 - 2
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[4,2]
=> [2]
=> []
=> []
=> ? = 3 - 2
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[3,3]
=> [3]
=> []
=> []
=> ? = 3 - 2
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[4,3]
=> [3]
=> []
=> []
=> ? = 3 - 2
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[5,5,2,1,1,1]
=> [5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[5,5,3,2,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[]
=> ?
=> ?
=> ?
=> ? = 0 - 2
[5,5,4,3,2,1]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[5,5,3,2,2,1]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[5,5,4,2,1,1]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St001657
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1 - 2
[2]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,1]
=> [1]
=> []
=> ?
=> ? = 2 - 2
[3]
=> []
=> ?
=> ?
=> ? = 3 - 2
[2,1]
=> [1]
=> []
=> ?
=> ? = 2 - 2
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[4]
=> []
=> ?
=> ?
=> ? = 3 - 2
[3,1]
=> [1]
=> []
=> ?
=> ? = 2 - 2
[2,2]
=> [2]
=> []
=> ?
=> ? = 3 - 2
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[4,1]
=> [1]
=> []
=> ?
=> ? = 3 - 2
[3,2]
=> [2]
=> []
=> ?
=> ? = 3 - 2
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[4,2]
=> [2]
=> []
=> ?
=> ? = 3 - 2
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[3,3]
=> [3]
=> []
=> ?
=> ? = 3 - 2
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[2,2,2]
=> [2,2]
=> [2]
=> [2]
=> 1 = 3 - 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[4,3]
=> [3]
=> []
=> ?
=> ? = 3 - 2
[4,2,1]
=> [2,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[3,3,1]
=> [3,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[3,2,2]
=> [2,2]
=> [2]
=> [2]
=> 1 = 3 - 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 0 = 2 - 2
[4,3,1]
=> [3,1]
=> [1]
=> [1]
=> 0 = 2 - 2
[4,2,2]
=> [2,2]
=> [2]
=> [2]
=> 1 = 3 - 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[3,3,2]
=> [3,2]
=> [2]
=> [2]
=> 1 = 3 - 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 0 = 2 - 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
[4,3,2]
=> [3,2]
=> [2]
=> [2]
=> 1 = 3 - 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0 = 2 - 2
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 0 = 2 - 2
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [3]
=> 0 = 2 - 2
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> [3]
=> 0 = 2 - 2
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 0 = 2 - 2
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 0 = 2 - 2
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [5,1]
=> 0 = 2 - 2
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 0 = 2 - 2
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 0 = 2 - 2
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [5,1]
=> 0 = 2 - 2
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [7]
=> 0 = 2 - 2
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 0 = 2 - 2
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> [5,1]
=> 0 = 2 - 2
[5,5,2,1,1,1]
=> [5,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 0 = 2 - 2
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [5,3]
=> 0 = 2 - 2
[5,5,3,2,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> [3,3,1]
=> 0 = 2 - 2
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [5,3]
=> 0 = 2 - 2
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [3,3,3]
=> 0 = 2 - 2
[]
=> ?
=> ?
=> ?
=> ? = 0 - 2
[5,5,4,3,2,1]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [5,5]
=> 0 = 2 - 2
[5,5,3,2,2,1]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [5,3]
=> 0 = 2 - 2
[5,5,4,2,1,1]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [6,1,1]
=> 0 = 2 - 2
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000646
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 75%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,4,2] => 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,1,4,3,2] => 1 = 2 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6,3,5,1,4,2] => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 2 = 3 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 1 = 2 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,6,5,1,4,3] => 2 = 3 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,5,2] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => 2 = 3 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 3 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 2 = 3 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [6,4,1,5,3,2] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,6,4,2] => 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 2 = 3 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 1 = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,5,4,2] => 1 = 2 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 3 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 1 = 2 - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 2 = 3 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 2 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 1 = 2 - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 2 = 3 - 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1 = 2 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 1 = 2 - 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1 = 2 - 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 1 = 2 - 1
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[5,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[4,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [7,1,2,3,6,4,8,5] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[]
=> []
=> [1] => [1] => ? = 0 - 1
[5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[5,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
[5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => ? = 2 - 1
Description
The number of big ascents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$.
For the number of small ascents, see [[St000441]].
Matching statistic: St000478
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ? = 1 - 2
[2]
=> []
=> ?
=> ? = 3 - 2
[1,1]
=> [1]
=> []
=> ? = 2 - 2
[3]
=> []
=> ?
=> ? = 3 - 2
[2,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4]
=> []
=> ?
=> ? = 3 - 2
[3,1]
=> [1]
=> []
=> ? = 2 - 2
[2,2]
=> [2]
=> []
=> ? = 3 - 2
[2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,1]
=> [1]
=> []
=> ? = 3 - 2
[3,2]
=> [2]
=> []
=> ? = 3 - 2
[3,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[2,2,1]
=> [2,1]
=> [1]
=> ? = 2 - 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,2]
=> [2]
=> []
=> ? = 3 - 2
[4,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[3,3]
=> [3]
=> []
=> ? = 3 - 2
[3,2,1]
=> [2,1]
=> [1]
=> ? = 2 - 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
[2,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,3]
=> [3]
=> []
=> ? = 3 - 2
[4,2,1]
=> [2,1]
=> [1]
=> ? = 2 - 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
[3,3,1]
=> [3,1]
=> [1]
=> ? = 2 - 2
[3,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[4,3,1]
=> [3,1]
=> [1]
=> ? = 2 - 2
[4,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
[3,3,2]
=> [3,2]
=> [2]
=> 1 = 3 - 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
[4,3,2]
=> [3,2]
=> [2]
=> 1 = 3 - 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> 0 = 2 - 2
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> 0 = 2 - 2
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 0 = 2 - 2
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 0 = 2 - 2
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> 0 = 2 - 2
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 0 = 2 - 2
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> 0 = 2 - 2
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> 0 = 2 - 2
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> 0 = 2 - 2
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> 0 = 2 - 2
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> 0 = 2 - 2
[5,5,2,1,1,1]
=> [5,2,1,1,1]
=> [2,1,1,1]
=> 0 = 2 - 2
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> 0 = 2 - 2
[5,5,3,2,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> 0 = 2 - 2
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> 0 = 2 - 2
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> 0 = 2 - 2
[]
=> ?
=> ?
=> ? = 0 - 2
[5,5,4,3,2,1]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> 0 = 2 - 2
[5,5,3,2,2,1]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> 0 = 2 - 2
[5,5,4,2,1,1]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> 0 = 2 - 2
Description
Another weight of a partition according to Alladi.
According to Theorem 3.4 (Alladi 2012) in [1]
$$
\sum_{\pi\in GG_1(r)} w_1(\pi)
$$
equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
Matching statistic: St000993
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1 - 1
[2]
=> []
=> ?
=> ?
=> ? = 3 - 1
[1,1]
=> [1]
=> []
=> []
=> ? = 2 - 1
[3]
=> []
=> ?
=> ?
=> ? = 3 - 1
[2,1]
=> [1]
=> []
=> []
=> ? = 2 - 1
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? = 2 - 1
[4]
=> []
=> ?
=> ?
=> ? = 3 - 1
[3,1]
=> [1]
=> []
=> []
=> ? = 2 - 1
[2,2]
=> [2]
=> []
=> []
=> ? = 3 - 1
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[4,1]
=> [1]
=> []
=> []
=> ? = 3 - 1
[3,2]
=> [2]
=> []
=> []
=> ? = 3 - 1
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? = 2 - 1
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[4,2]
=> [2]
=> []
=> []
=> ? = 3 - 1
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? = 2 - 1
[3,3]
=> [3]
=> []
=> []
=> ? = 3 - 1
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[4,3]
=> [3]
=> []
=> []
=> ? = 3 - 1
[4,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[3,3,1]
=> [3,1]
=> [1]
=> [1]
=> ? = 2 - 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[4,3,1]
=> [3,1]
=> [1]
=> [1]
=> ? = 2 - 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[4,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 1 = 2 - 1
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 1 = 2 - 1
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [4,3]
=> 1 = 2 - 1
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 1 = 2 - 1
[5,5,2,1,1,1]
=> [5,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [4,3,1]
=> 1 = 2 - 1
[5,5,3,2,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> [4,2,1]
=> 1 = 2 - 1
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [4,3,1]
=> 1 = 2 - 1
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [4,3,2]
=> 1 = 2 - 1
[]
=> ?
=> ?
=> ?
=> ? = 0 - 1
[5,5,4,3,2,1]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [4,3,2,1]
=> 1 = 2 - 1
[5,5,3,2,2,1]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [4,3,1]
=> 1 = 2 - 1
[5,5,4,2,1,1]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [4,2,1,1]
=> 1 = 2 - 1
Description
The multiplicity of the largest part of an integer partition.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000929The constant term of the character polynomial of an integer partition. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000264The girth of a graph, which is not a tree. St000259The diameter of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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