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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000353
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000023
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 1
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1
[8,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0
[4,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1
[9,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0
[6,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,6,3,4,7] => ? = 1
[5,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 1
[5,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1
[3,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 1
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 1
[10,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0
[7,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,6,3,4,7] => ? = 1
[6,6,4]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,2,3,6,4,7] => ? = 1
[6,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 1
[6,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1
[5,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1
[4,4,4,4]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 1
[4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,7,4,5] => ? = 2
[4,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 1
[4,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 1
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 1
[11,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0
[8,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,6,3,4,7] => ? = 1
[7,6,4]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,2,3,6,4,7] => ? = 1
[7,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 1
[7,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1
[6,6,5]
=> [6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 0
[6,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1
[5,4,4,4]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 1
[5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,7,4,5] => ? = 2
[5,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 1
[5,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 1
[4,4,3,3,3]
=> [4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,1,2,6,7,4,5] => ? = 1
[4,4,3,2,2,2]
=> [4,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,5,6,7,3,4] => ? = 1
[4,3,3,3,2,2]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 1
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 1
[7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 0
[6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 0
[7,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1
[6,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1
[5,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1
[7,6,4,4,3,2]
=> [6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 1
[6,5,4,4,3,2]
=> [5,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 1
[6,5,4,3,3,2]
=> [5,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => ? = 1
[5,4,3,3,3,2]
=> [4,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,4,5] => ? = 1
[4,3,3,3,3,2]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000092
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 0 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2 = 1 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2 = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 2 = 1 + 1
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 0 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2 = 1 + 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2 = 1 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2 = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 2 = 1 + 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 0 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1 = 0 + 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2 = 1 + 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2 = 1 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2 = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 1 + 1
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 0 + 1
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? = 1 + 1
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 0 + 1
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? = 1 + 1
[8,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 0 + 1
[4,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? = 1 + 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,7,2,4,5,6] => ? = 1 + 1
[9,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 0 + 1
[6,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,2,6,3,4,7,5] => ? = 1 + 1
[5,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => ? = 1 + 1
[5,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? = 1 + 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,7,2,4,5,6] => ? = 1 + 1
[3,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ? = 1 + 1
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => ? = 1 + 1
[10,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 0 + 1
[7,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,2,6,3,4,7,5] => ? = 1 + 1
[6,6,4]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5] => ? = 1 + 1
[6,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => ? = 1 + 1
[6,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? = 1 + 1
[5,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,7,2,4,5,6] => ? = 1 + 1
[4,4,4,4]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => ? = 1 + 1
[4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,3,2,7,4,5,6] => ? = 2 + 1
[4,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ? = 1 + 1
[4,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => ? = 1 + 1
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => ? = 1 + 1
[11,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 0 + 1
[8,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,2,6,3,4,7,5] => ? = 1 + 1
[7,6,4]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5] => ? = 1 + 1
[7,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => ? = 1 + 1
[7,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? = 1 + 1
[6,6,5]
=> [6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => ? = 0 + 1
[6,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,7,2,4,5,6] => ? = 1 + 1
[5,4,4,4]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => ? = 1 + 1
[5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,3,2,7,4,5,6] => ? = 2 + 1
[5,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ? = 1 + 1
[5,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => ? = 1 + 1
[4,4,3,3,3]
=> [4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,2,4,7,3,5,6] => ? = 1 + 1
[4,4,3,2,2,2]
=> [4,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,6,7,2,4,5] => ? = 1 + 1
[4,3,3,3,2,2]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => ? = 1 + 1
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ? = 1 + 1
[7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => ? = 0 + 1
[6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => ? = 0 + 1
[7,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => ? = 1 + 1
[6,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => ? = 1 + 1
[5,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => ? = 1 + 1
[7,6,4,4,3,2]
=> [6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,2,7,6] => ? = 1 + 1
[6,5,4,4,3,2]
=> [5,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,3,4,5,7,2,6] => ? = 1 + 1
[6,5,4,3,3,2]
=> [5,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,6,7,2,5] => ? = 1 + 1
[5,4,3,3,3,2]
=> [4,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,3,4,7,2,5,6] => ? = 1 + 1
[4,3,3,3,3,2]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ? = 1 + 1
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000099
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2 = 1 + 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 0 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 2 = 1 + 1
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 0 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 1 + 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2 = 1 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 2 = 1 + 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 0 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 1 + 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1 = 0 + 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 1 + 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2 = 1 + 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0 + 1
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1 + 1
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0 + 1
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1 + 1
[8,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0 + 1
[4,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1 + 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1 + 1
[9,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0 + 1
[6,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,6,3,4,7] => ? = 1 + 1
[5,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 1 + 1
[5,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1 + 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1 + 1
[3,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 1 + 1
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 1 + 1
[10,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0 + 1
[7,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,6,3,4,7] => ? = 1 + 1
[6,6,4]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,2,3,6,4,7] => ? = 1 + 1
[6,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 1 + 1
[6,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1 + 1
[5,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1 + 1
[4,4,4,4]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 1 + 1
[4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,7,4,5] => ? = 2 + 1
[4,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 1 + 1
[4,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 1 + 1
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 1 + 1
[11,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 0 + 1
[8,6,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,2,6,3,4,7] => ? = 1 + 1
[7,6,4]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,2,3,6,4,7] => ? = 1 + 1
[7,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 1 + 1
[7,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 1 + 1
[6,6,5]
=> [6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 0 + 1
[6,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => ? = 1 + 1
[5,4,4,4]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 1 + 1
[5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,7,4,5] => ? = 2 + 1
[5,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 1 + 1
[5,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 1 + 1
[4,4,3,3,3]
=> [4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,1,2,6,7,4,5] => ? = 1 + 1
[4,4,3,2,2,2]
=> [4,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,5,6,7,3,4] => ? = 1 + 1
[4,3,3,3,2,2]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 1 + 1
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 1 + 1
[7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 0 + 1
[6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 0 + 1
[7,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1 + 1
[6,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1 + 1
[5,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1 + 1
[7,6,4,4,3,2]
=> [6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 1 + 1
[6,5,4,4,3,2]
=> [5,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 1 + 1
[6,5,4,3,3,2]
=> [5,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => ? = 1 + 1
[5,4,3,3,3,2]
=> [4,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,4,5] => ? = 1 + 1
[4,3,3,3,3,2]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 1 + 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St001960
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[6,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[8,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 0
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 2
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[6,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[5,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0
[5,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[4,4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
[4,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 2
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ? = 2
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ? = 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 2
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? = 1
[7,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[6,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0
[6,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000461
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 0
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => 0
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,2,3,4] => 0
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5,1,2,4,3] => 0
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,1,2,5,3] => 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,1,5,3,2] => 0
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [3,1,4,5,2] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,3,4,5,1] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 0
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => 0
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [5,1,2,3,6,4] => 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => 0
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [5,1,2,6,4,3] => 0
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [4,1,2,5,6,3] => 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,1,6,3,5,2] => 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [4,1,5,3,6,2] => 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [3,1,4,5,6,2] => 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [2,3,4,5,6,1] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 0
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => ? = 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [6,1,2,3,7,5,4] => ? = 0
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [5,1,2,3,6,7,4] => ? = 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [6,1,2,7,4,5,3] => ? = 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [5,1,2,7,4,6,3] => ? = 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [5,1,2,6,4,7,3] => ? = 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [4,1,2,5,6,7,3] => ? = 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [4,1,6,3,7,5,2] => ? = 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [4,1,5,3,6,7,2] => ? = 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [3,1,4,5,6,7,2] => ? = 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => ? = 0
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [8,1,2,3,4,5,7,6] => ? = 0
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,8,6] => ? = 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [8,1,2,3,4,6,7,5] => ? = 0
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [7,1,2,3,4,8,6,5] => ? = 0
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [6,1,2,3,4,7,8,5] => ? = 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [8,1,2,3,5,6,7,4] => ? = 0
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [7,1,2,3,8,5,6,4] => ? = 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [6,1,2,3,8,5,7,4] => ? = 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [6,1,2,3,7,5,8,4] => ? = 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [5,1,2,3,6,7,8,4] => ? = 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [6,1,2,8,4,5,7,3] => ? = 0
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [6,1,2,7,4,5,8,3] => ? = 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [5,1,2,7,4,8,6,3] => ? = 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [5,1,2,6,4,7,8,3] => ? = 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [4,1,2,5,6,7,8,3] => ? = 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,7,2] => ? = 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [4,1,6,3,7,5,8,2] => ? = 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [4,1,5,3,6,7,8,2] => ? = 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => ? = 0
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [9,1,2,3,4,5,6,8,7] => ? = 0
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,9,7] => ? = 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [9,1,2,3,4,5,7,8,6] => ? = 0
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [8,1,2,3,4,5,9,7,6] => ? = 0
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,8,9,6] => ? = 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [9,1,2,3,4,6,7,8,5] => ? = 0
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [8,1,2,3,4,9,6,7,5] => ? = 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [7,1,2,3,4,9,6,8,5] => ? = 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [7,1,2,3,4,8,6,9,5] => ? = 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [6,1,2,3,4,7,8,9,5] => ? = 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [8,1,2,3,9,5,6,7,4] => ? = 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [7,1,2,3,9,5,6,8,4] => ? = 0
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [7,1,2,3,8,5,6,9,4] => ? = 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [6,1,2,3,8,5,9,7,4] => ? = 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [6,1,2,3,7,5,8,9,4] => ? = 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] => [5,1,2,3,6,7,8,9,4] => ? = 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [6,1,2,9,4,5,7,8,3] => ? = 1
Description
The rix statistic of a permutation.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
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