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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St000531
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,2,2]
=> [2,2]
=> [2]
=> [2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 4
[5,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[4,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,3,1]
=> [3,1]
=> [1]
=> [1]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 4
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[6,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[5,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[4,3,1]
=> [3,1]
=> [1]
=> [1]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[3,3,2]
=> [3,2]
=> [2]
=> [2]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 4
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [4]
=> 4
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 4
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 6
[7,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[6,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 2
[5,3,1]
=> [3,1]
=> [1]
=> [1]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [2]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 3
[4,4,1]
=> [4,1]
=> [1]
=> [1]
=> 1
Description
The leading coefficient of the rook polynomial of an integer partition.
Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
Matching statistic: St001232
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 26%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 26%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[4,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[4,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2
[3,3,3,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 2
[3,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 6
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[6,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[6,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3
[5,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[5,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[5,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4
[5,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2
[4,4,3]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[4,4,2,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 3
[4,3,3,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 2
[4,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[4,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Values
[1,1,1]
=> [1,1]
=> 110 => 110 => 1
[2,1,1]
=> [1,1]
=> 110 => 110 => 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[3,1,1]
=> [1,1]
=> 110 => 110 => 1
[2,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? = 3
[4,1,1]
=> [1,1]
=> 110 => 110 => 1
[3,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[2,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[2,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? = 2
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? = 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => ? = 4
[5,1,1]
=> [1,1]
=> 110 => 110 => 1
[4,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[3,3,1]
=> [3,1]
=> 10010 => 10100 => ? = 1
[3,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[3,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? = 2
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? = 3
[2,2,2,1]
=> [2,2,1]
=> 11010 => 11100 => ? = 3
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 110110 => ? = 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => ? = 4
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => ? = 5
[6,1,1]
=> [1,1]
=> 110 => 110 => 1
[5,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[4,3,1]
=> [3,1]
=> 10010 => 10100 => ? = 1
[4,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[4,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? = 2
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? = 3
[3,3,2]
=> [3,2]
=> 10100 => 01100 => ? = 2
[3,3,1,1]
=> [3,1,1]
=> 100110 => 101010 => ? = 2
[3,2,2,1]
=> [2,2,1]
=> 11010 => 11100 => ? = 3
[3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 110110 => ? = 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => ? = 4
[2,2,2,2]
=> [2,2,2]
=> 11100 => 01110 => ? = 4
[2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 111010 => ? = 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 1101110 => ? = 4
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => ? = 5
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 11111110 => ? = 6
[7,1,1]
=> [1,1]
=> 110 => 110 => 1
[6,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[6,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[5,3,1]
=> [3,1]
=> 10010 => 10100 => ? = 1
[5,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[5,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? = 2
[5,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? = 3
[4,4,1]
=> [4,1]
=> 100010 => 100100 => ? = 1
[4,3,2]
=> [3,2]
=> 10100 => 01100 => ? = 2
[4,3,1,1]
=> [3,1,1]
=> 100110 => 101010 => ? = 2
[4,2,2,1]
=> [2,2,1]
=> 11010 => 11100 => ? = 3
[4,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 110110 => ? = 3
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => ? = 4
[3,3,3]
=> [3,3]
=> 11000 => 00110 => ? = 1
[3,3,2,1]
=> [3,2,1]
=> 101010 => 111000 => ? = 3
[3,3,1,1,1]
=> [3,1,1,1]
=> 1001110 => 1010110 => ? = 3
[3,2,2,2]
=> [2,2,2]
=> 11100 => 01110 => ? = 4
[3,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 111010 => ? = 2
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 1101110 => ? = 4
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => ? = 5
[2,2,2,2,1]
=> [2,2,2,1]
=> 111010 => 111100 => ? = 5
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> 1101110 => 1110110 => ? = 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> 10111110 => 11011110 => ? = 5
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 11111110 => ? = 6
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 111111110 => ? = 7
[8,1,1]
=> [1,1]
=> 110 => 110 => 1
[7,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[7,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[6,3,1]
=> [3,1]
=> 10010 => 10100 => ? = 1
[6,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[6,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? = 2
[6,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? = 3
[5,4,1]
=> [4,1]
=> 100010 => 100100 => ? = 1
[5,3,2]
=> [3,2]
=> 10100 => 01100 => ? = 2
[9,1,1]
=> [1,1]
=> 110 => 110 => 1
[8,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[8,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[7,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[10,1,1]
=> [1,1]
=> 110 => 110 => 1
[9,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[9,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[8,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[11,1,1]
=> [1,1]
=> 110 => 110 => 1
[10,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[10,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[9,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[12,1,1]
=> [1,1]
=> 110 => 110 => 1
[11,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[11,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[10,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[13,1,1]
=> [1,1]
=> 110 => 110 => 1
[12,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[12,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[11,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[14,1,1]
=> [1,1]
=> 110 => 110 => 1
[13,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[13,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[12,2,2]
=> [2,2]
=> 1100 => 0110 => 2
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001207
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 3 + 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 3 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 4 + 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 3 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 3 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 3 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 4 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 3 + 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 3 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 3 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 4 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 4 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 2 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 4 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => ? = 6 + 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 3 + 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1 + 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 + 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 3 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 3 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 4 + 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ? = 1 + 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 3 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 3 + 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 4 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 2 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 4 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ? = 5 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ? = 0 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ? = 5 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => ? = 6 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => ? = 7 + 1
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[7,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[6,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 3 + 1
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1 + 1
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 1
[9,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[8,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[10,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[9,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[11,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[10,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[10,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[9,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[12,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[11,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[11,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[10,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[13,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[12,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[12,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[11,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[14,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[13,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[13,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[12,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001087
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 35%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 35%
Values
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 3
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 3
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 4
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ? = 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 2
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ? = 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? = 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => ? = 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => 3
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => ? = 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 4
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 5
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,6,8],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8] => ? = 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => 2
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,5,8],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8] => ? = 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,5,7],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7] => ? = 2
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,5,7,8],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8] => ? = 2
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => 3
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,4,7],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7] => ? = 2
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => ? = 2
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,4,6,8],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8] => ? = 3
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[1,4,6,7,8],[2,5],[3]]
=> [3,2,5,1,4,6,7,8] => ? = 3
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => 4
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => 4
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => ? = 2
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => ? = 4
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => 5
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 6
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [[1,7,9],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9] => ? = 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => 2
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,6,9],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6,9] => ? = 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,6,8],[2,7,9],[3],[4],[5]]
=> [5,4,3,2,7,9,1,6,8] => ? = 2
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [[1,6,8,9],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8,9] => ? = 2
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => 3
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5,9] => ? = 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> [4,3,7,2,6,9,1,5,8] => ? = 2
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8,9] => ? = 2
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7,9] => ? = 3
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8,9] => ? = 3
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => 4
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> [3,6,9,2,5,8,1,4,7] => ? = 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7,9] => ? = 3
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8,9] => ? = 3
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> [3,2,5,7,9,1,4,6,8] => ? = 4
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8,9] => ? = 2
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> [3,2,5,1,4,6,7,8,9] => ? = 4
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => 5
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7,9] => ? = 5
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> [2,4,6,1,3,5,7,8,9] => ? = 0
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,3,5,6,7,8,9],[2,4]]
=> [2,4,1,3,5,6,7,8,9] => ? = 5
[2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => 6
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => 7
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9,10] => 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [[1,8,10],[2,9],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,9,1,8,10] => ? = 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9,10] => 2
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [[1,7,10],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7,10] => ? = 1
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [[1,7,9],[2,8,10],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,10,1,7,9] => ? = 2
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [[1,7,9,10],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9,10] => ? = 2
[6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9,10] => 3
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [[1,6,10],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6,10] => ? = 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [[1,6,9],[2,7,10],[3,8],[4],[5]]
=> [5,4,3,8,2,7,10,1,6,9] => ? = 2
[5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [[1,6,9,10],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6,9,10] => ? = 2
[5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [[1,6,8,10],[2,7,9],[3],[4],[5]]
=> [5,4,3,2,7,9,1,6,8,10] => ? = 3
[5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [[1,6,8,9,10],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8,9,10] => ? = 3
[5,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10]]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => 4
[4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [[1,5,9],[2,6,10],[3,7],[4,8]]
=> [4,8,3,7,2,6,10,1,5,9] => ? = 2
[4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [[1,5,9,10],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5,9,10] => ? = 2
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [[1,5,8],[2,6,9],[3,7,10],[4]]
=> [4,3,7,10,2,6,9,1,5,8] => ? = 1
[4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [[1,5,8,10],[2,6,9],[3,7],[4]]
=> [4,3,7,2,6,9,1,5,8,10] => ? = 3
[4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [[1,5,8,9,10],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8,9,10] => ? = 3
[4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [[1,5,7,9],[2,6,8,10],[3],[4]]
=> [4,3,2,6,8,10,1,5,7,9] => ? = 4
[4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [[1,5,7,9,10],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7,9,10] => ? = 2
[4,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [[1,5,7,8,9,10],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8,9,10] => ? = 4
[4,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10]]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => 5
[3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => 6
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => 6
[2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => 7
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => 8
[2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => 8
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: St000366
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 39%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 39%
Values
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 3
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,4,2,6,5,3] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 3
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 4
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 2
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [4,1,5,2,7,6,3] => ? = 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,4,2,7,5,3] => ? = 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 3
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [5,3,1,7,6,4,2] => ? = 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [3,1,7,6,5,4,2] => ? = 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 4
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 5
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [1,2,3,4,5,8,7,6] => ? = 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,6,4,8,7,5] => ? = 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,4,8,7,6,5] => ? = 2
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,5,2,6,3,8,7,4] => ? = 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,2,7,5,3,8,6,4] => ? = 2
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,5,3,8,7,6,4] => ? = 2
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [1,2,3,8,7,6,5,4] => ? = 3
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [4,1,7,5,2,8,6,3] => ? = 2
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [4,1,5,2,8,7,6,3] => ? = 2
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,6,4,2,8,7,5,3] => ? = 3
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,4,2,8,7,6,5,3] => ? = 3
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,2,8,7,6,5,4,3] => ? = 4
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 4
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [5,3,1,8,7,6,4,2] => ? = 2
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [3,1,8,7,6,5,4,2] => ? = 4
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => 5
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 6
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,8,7] => ? = 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,9,8,6] => ? = 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,9,8,7,6] => ? = 2
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,9,8,5] => ? = 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [1,2,3,8,6,4,9,7,5] => ? = 2
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [1,2,3,6,4,9,8,7,5] => ? = 2
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => ? = 3
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,9,8,4] => ? = 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [1,5,2,8,6,3,9,7,4] => ? = 2
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [1,5,2,6,3,9,8,7,4] => ? = 2
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [1,2,7,5,3,9,8,6,4] => ? = 3
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [1,2,5,3,9,8,7,6,4] => ? = 3
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => [1,2,3,9,8,7,6,5,4] => ? = 4
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [7,4,1,8,5,2,9,6,3] => ? = 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [4,1,7,5,2,9,8,6,3] => ? = 3
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [4,1,5,2,9,8,7,6,3] => ? = 3
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [1,8,6,4,2,9,7,5,3] => ? = 4
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [1,6,4,2,9,8,7,5,3] => ? = 2
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => [1,4,2,9,8,7,6,5,3] => ? = 4
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,1,2,3] => [1,2,9,8,7,6,5,4,3] => ? = 5
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [7,5,3,1,9,8,6,4,2] => ? = 5
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => [5,3,1,9,8,7,6,4,2] => ? = 0
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => [3,1,9,8,7,6,5,4,2] => ? = 5
[2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,1,2] => [1,9,8,7,6,5,4,3,2] => 6
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 7
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [10,9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,10,9,8] => ? = 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [10,8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,8,6,10,9,7] => ? = 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [10,9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,10,9,8,7] => ? = 2
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [10,7,8,9,1,2,3,4,5,6] => [1,2,3,7,4,8,5,10,9,6] => ? = 1
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [9,10,7,8,1,2,3,4,5,6] => [1,2,3,4,9,7,5,10,8,6] => ? = 2
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [10,9,7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,10,9,8,6] => ? = 2
[6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [10,9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,10,9,8,7,6] => ? = 3
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [10,6,7,8,9,1,2,3,4,5] => [1,6,2,7,3,8,4,10,9,5] => ? = 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [9,10,6,7,8,1,2,3,4,5] => [1,2,6,3,9,7,4,10,8,5] => ? = 2
[2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,4,3,1,2] => [1,10,9,8,7,6,5,4,3,2] => 7
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 8
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 10
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000371
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 39%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 39%
Values
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 3
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,4,2,6,5,3] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 3
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 4
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 2
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [4,1,5,2,7,6,3] => ? = 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,4,2,7,5,3] => ? = 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 3
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [5,3,1,7,6,4,2] => ? = 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [3,1,7,6,5,4,2] => ? = 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 4
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 5
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [1,2,3,4,5,8,7,6] => ? = 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,6,4,8,7,5] => ? = 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,4,8,7,6,5] => ? = 2
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,5,2,6,3,8,7,4] => ? = 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,2,7,5,3,8,6,4] => ? = 2
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,5,3,8,7,6,4] => ? = 2
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [1,2,3,8,7,6,5,4] => ? = 3
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [4,1,7,5,2,8,6,3] => ? = 2
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [4,1,5,2,8,7,6,3] => ? = 2
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,6,4,2,8,7,5,3] => ? = 3
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,4,2,8,7,6,5,3] => ? = 3
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,2,8,7,6,5,4,3] => ? = 4
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 4
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [5,3,1,8,7,6,4,2] => ? = 2
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [3,1,8,7,6,5,4,2] => ? = 4
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => 5
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 6
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,8,7] => ? = 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,9,8,6] => ? = 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,9,8,7,6] => ? = 2
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,9,8,5] => ? = 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [1,2,3,8,6,4,9,7,5] => ? = 2
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [1,2,3,6,4,9,8,7,5] => ? = 2
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => ? = 3
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,9,8,4] => ? = 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [1,5,2,8,6,3,9,7,4] => ? = 2
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [1,5,2,6,3,9,8,7,4] => ? = 2
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [1,2,7,5,3,9,8,6,4] => ? = 3
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [1,2,5,3,9,8,7,6,4] => ? = 3
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => [1,2,3,9,8,7,6,5,4] => ? = 4
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [7,4,1,8,5,2,9,6,3] => ? = 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [4,1,7,5,2,9,8,6,3] => ? = 3
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [4,1,5,2,9,8,7,6,3] => ? = 3
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [1,8,6,4,2,9,7,5,3] => ? = 4
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [1,6,4,2,9,8,7,5,3] => ? = 2
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => [1,4,2,9,8,7,6,5,3] => ? = 4
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,1,2,3] => [1,2,9,8,7,6,5,4,3] => ? = 5
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [7,5,3,1,9,8,6,4,2] => ? = 5
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => [5,3,1,9,8,7,6,4,2] => ? = 0
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => [3,1,9,8,7,6,5,4,2] => ? = 5
[2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,1,2] => [1,9,8,7,6,5,4,3,2] => 6
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 7
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [10,9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,10,9,8] => ? = 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [10,8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,8,6,10,9,7] => ? = 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [10,9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,10,9,8,7] => ? = 2
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [10,7,8,9,1,2,3,4,5,6] => [1,2,3,7,4,8,5,10,9,6] => ? = 1
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [9,10,7,8,1,2,3,4,5,6] => [1,2,3,4,9,7,5,10,8,6] => ? = 2
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [10,9,7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,10,9,8,6] => ? = 2
[6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [10,9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,10,9,8,7,6] => ? = 3
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [10,6,7,8,9,1,2,3,4,5] => [1,6,2,7,3,8,4,10,9,5] => ? = 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [9,10,6,7,8,1,2,3,4,5] => [1,2,6,3,9,7,4,10,8,5] => ? = 2
[2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,4,3,1,2] => [1,10,9,8,7,6,5,4,3,2] => 7
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 8
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 10
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000782
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 4%
Values
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 3
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 4
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 3
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 4
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 3
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 4
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 4
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 4
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 6
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 1
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 3
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> ? = 1
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 2
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 3
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 4
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> ? = 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 3
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> ? = 3
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 4
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 2
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 4
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[11,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[12,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[13,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[14,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[13,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[15,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[14,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001722
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 4%
Values
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 4
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 4
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 5
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 4
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 4
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 4
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 5
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 6
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 1
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 2
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 3
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 4
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 3
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 3
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 4
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 2
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 4
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 5
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[11,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[12,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[13,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[14,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[13,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[15,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[14,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000662
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 35%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 35%
Values
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 2 = 1 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 3 = 2 + 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 2 = 1 + 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => 2 = 1 + 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 3 = 2 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4 = 3 + 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 2 = 1 + 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,4,2,6,5,3] => 2 = 1 + 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 3 = 2 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => 3 = 2 + 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => 3 = 2 + 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 4 = 3 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5 = 4 + 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 2 = 1 + 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => 2 = 1 + 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 3 = 2 + 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [4,1,5,2,7,6,3] => ? = 1 + 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,4,2,7,5,3] => ? = 2 + 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => 3 = 2 + 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 4 = 3 + 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [5,3,1,7,6,4,2] => ? = 3 + 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [3,1,7,6,5,4,2] => ? = 3 + 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 5 = 4 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 6 = 5 + 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [1,2,3,4,5,8,7,6] => ? = 1 + 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,6,4,8,7,5] => ? = 1 + 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,4,8,7,6,5] => ? = 2 + 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,5,2,6,3,8,7,4] => ? = 1 + 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,2,7,5,3,8,6,4] => ? = 2 + 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,5,3,8,7,6,4] => ? = 2 + 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [1,2,3,8,7,6,5,4] => ? = 3 + 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [4,1,7,5,2,8,6,3] => ? = 2 + 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [4,1,5,2,8,7,6,3] => ? = 2 + 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,6,4,2,8,7,5,3] => ? = 3 + 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,4,2,8,7,6,5,3] => ? = 3 + 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,2,8,7,6,5,4,3] => ? = 4 + 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 5 = 4 + 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [5,3,1,8,7,6,4,2] => ? = 2 + 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [3,1,8,7,6,5,4,2] => ? = 4 + 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => 6 = 5 + 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 7 = 6 + 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,8,7] => ? = 1 + 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,9,8,6] => ? = 1 + 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,9,8,7,6] => ? = 2 + 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,9,8,5] => ? = 1 + 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [1,2,3,8,6,4,9,7,5] => ? = 2 + 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [1,2,3,6,4,9,8,7,5] => ? = 2 + 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => ? = 3 + 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,9,8,4] => ? = 1 + 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [1,5,2,8,6,3,9,7,4] => ? = 2 + 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [1,5,2,6,3,9,8,7,4] => ? = 2 + 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [1,2,7,5,3,9,8,6,4] => ? = 3 + 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [1,2,5,3,9,8,7,6,4] => ? = 3 + 1
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => [1,2,3,9,8,7,6,5,4] => ? = 4 + 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [7,4,1,8,5,2,9,6,3] => ? = 1 + 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [4,1,7,5,2,9,8,6,3] => ? = 3 + 1
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [4,1,5,2,9,8,7,6,3] => ? = 3 + 1
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [1,8,6,4,2,9,7,5,3] => ? = 4 + 1
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [1,6,4,2,9,8,7,5,3] => ? = 2 + 1
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => [1,4,2,9,8,7,6,5,3] => ? = 4 + 1
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,1,2,3] => [1,2,9,8,7,6,5,4,3] => ? = 5 + 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [7,5,3,1,9,8,6,4,2] => ? = 5 + 1
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => [5,3,1,9,8,7,6,4,2] => ? = 0 + 1
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => [3,1,9,8,7,6,5,4,2] => ? = 5 + 1
[2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,1,2] => [1,9,8,7,6,5,4,3,2] => 7 = 6 + 1
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 8 = 7 + 1
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [10,9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,10,9,8] => ? = 1 + 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [10,8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,8,6,10,9,7] => ? = 1 + 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [10,9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,10,9,8,7] => ? = 2 + 1
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [10,7,8,9,1,2,3,4,5,6] => [1,2,3,7,4,8,5,10,9,6] => ? = 1 + 1
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [9,10,7,8,1,2,3,4,5,6] => [1,2,3,4,9,7,5,10,8,6] => ? = 2 + 1
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [10,9,7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,10,9,8,6] => ? = 2 + 1
[6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [10,9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,10,9,8,7,6] => ? = 3 + 1
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [10,6,7,8,9,1,2,3,4,5] => [1,6,2,7,3,8,4,10,9,5] => ? = 1 + 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [9,10,6,7,8,1,2,3,4,5] => [1,2,6,3,9,7,4,10,8,5] => ? = 2 + 1
[2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,4,3,1,2] => [1,10,9,8,7,6,5,4,3,2] => 8 = 7 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 9 = 8 + 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001816Eigenvalues of the top-to-random operator acting on a simple module. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000637The length of the longest cycle in a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000836The number of descents of distance 2 of a permutation. St000356The number of occurrences of the pattern 13-2. St000651The maximal size of a rise in a permutation.
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