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Your data matches 270 different statistics following compositions of up to 3 maps.
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Matching statistic: St001947
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(load all 3 compositions to match this statistic)
St001947: Parking functions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 1
[1,2] => 0
[2,1] => 0
[1,1,1] => 2
[1,1,2] => 1
[1,2,1] => 0
[2,1,1] => 1
[1,1,3] => 1
[1,3,1] => 0
[3,1,1] => 1
[1,2,2] => 1
[2,1,2] => 0
[2,2,1] => 1
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,1,1,1] => 3
[1,1,1,2] => 2
[1,1,2,1] => 1
[1,2,1,1] => 1
[2,1,1,1] => 2
[1,1,1,3] => 2
[1,1,3,1] => 1
[1,3,1,1] => 1
[3,1,1,1] => 2
[1,1,1,4] => 2
[1,1,4,1] => 1
[1,4,1,1] => 1
[4,1,1,1] => 2
[1,1,2,2] => 2
[1,2,1,2] => 0
[1,2,2,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 0
[2,2,1,1] => 2
[1,1,2,3] => 1
[1,1,3,2] => 1
[1,2,1,3] => 0
[1,2,3,1] => 0
[1,3,1,2] => 0
[1,3,2,1] => 0
[2,1,1,3] => 1
[2,1,3,1] => 0
[2,3,1,1] => 1
[3,1,1,2] => 1
[3,1,2,1] => 0
Description
The number of ties in a parking function.
This is the number of indices $i$ such that $p_i=p_{i+1}$.
Matching statistic: St000007
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00053: Parking functions āto car permutationā¶ Permutations
St000007: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000007: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,1] => [1,2] => 1 = 0 + 1
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => 2 = 1 + 1
[1,1,1] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [1,2,3] => 1 = 0 + 1
[1,2,1] => [1,2,3] => 1 = 0 + 1
[2,1,1] => [2,1,3] => 1 = 0 + 1
[1,1,3] => [1,2,3] => 1 = 0 + 1
[1,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [2,3,1] => 2 = 1 + 1
[1,2,2] => [1,2,3] => 1 = 0 + 1
[2,1,2] => [2,1,3] => 1 = 0 + 1
[2,2,1] => [3,1,2] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 1 = 0 + 1
[2,3,1] => [3,1,2] => 2 = 1 + 1
[3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [3,2,1] => 3 = 2 + 1
[1,1,1,1] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,2] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,1] => [1,2,3,4] => 1 = 0 + 1
[2,1,1,1] => [2,1,3,4] => 1 = 0 + 1
[1,1,1,3] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,1] => [1,2,3,4] => 1 = 0 + 1
[1,3,1,1] => [1,3,2,4] => 1 = 0 + 1
[3,1,1,1] => [2,3,1,4] => 1 = 0 + 1
[1,1,1,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,2] => [1,2,3,4] => 1 = 0 + 1
[1,2,2,1] => [1,2,3,4] => 1 = 0 + 1
[2,1,1,2] => [2,1,3,4] => 1 = 0 + 1
[2,1,2,1] => [2,1,3,4] => 1 = 0 + 1
[2,2,1,1] => [3,1,2,4] => 1 = 0 + 1
[1,1,2,3] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,2] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,3] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,1] => [1,2,3,4] => 1 = 0 + 1
[1,3,1,2] => [1,3,2,4] => 1 = 0 + 1
[1,3,2,1] => [1,3,2,4] => 1 = 0 + 1
[2,1,1,3] => [2,1,3,4] => 1 = 0 + 1
[2,1,3,1] => [2,1,3,4] => 1 = 0 + 1
[2,3,1,1] => [3,1,2,4] => 1 = 0 + 1
[3,1,1,2] => [2,3,1,4] => 1 = 0 + 1
[3,1,2,1] => [2,3,1,4] => 1 = 0 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000025
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00056: Parking functions āto Dyck pathā¶ Dyck paths
St000025: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000025: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 0 + 1
[1,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,0,1,0]
=> 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000084
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00297: Parking functions āordered treeā¶ Ordered trees
St000084: Ordered trees ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000084: Ordered trees ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [[]]
=> 1 = 0 + 1
[1,1] => [[],[]]
=> 2 = 1 + 1
[1,2] => [[[]]]
=> 1 = 0 + 1
[2,1] => [[[]]]
=> 1 = 0 + 1
[1,1,1] => [[],[],[]]
=> 3 = 2 + 1
[1,1,2] => [[],[[]]]
=> 2 = 1 + 1
[1,2,1] => [[[[]]]]
=> 1 = 0 + 1
[2,1,1] => [[],[[]]]
=> 2 = 1 + 1
[1,1,3] => [[],[[]]]
=> 2 = 1 + 1
[1,3,1] => [[[],[]]]
=> 1 = 0 + 1
[3,1,1] => [[],[[]]]
=> 2 = 1 + 1
[1,2,2] => [[],[[]]]
=> 2 = 1 + 1
[2,1,2] => [[[[]]]]
=> 1 = 0 + 1
[2,2,1] => [[],[[]]]
=> 2 = 1 + 1
[1,2,3] => [[[],[]]]
=> 1 = 0 + 1
[1,3,2] => [[[[]]]]
=> 1 = 0 + 1
[2,1,3] => [[[[]]]]
=> 1 = 0 + 1
[2,3,1] => [[[[]]]]
=> 1 = 0 + 1
[3,1,2] => [[[[]]]]
=> 1 = 0 + 1
[3,2,1] => [[[],[]]]
=> 1 = 0 + 1
[1,1,1,1] => [[],[],[],[]]
=> 4 = 3 + 1
[1,1,1,2] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,2,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,1,3] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,3,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,3,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[3,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,1,4] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,4,1] => [[],[[[]]]]
=> 2 = 1 + 1
[1,4,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[4,1,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,2] => [[],[],[[]]]
=> 3 = 2 + 1
[1,2,1,2] => [[[],[[]]]]
=> 1 = 0 + 1
[1,2,2,1] => [[[]],[[]]]
=> 2 = 1 + 1
[2,1,1,2] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,2,1] => [[[],[[]]]]
=> 1 = 0 + 1
[2,2,1,1] => [[],[],[[]]]
=> 3 = 2 + 1
[1,1,2,3] => [[],[[],[]]]
=> 2 = 1 + 1
[1,1,3,2] => [[],[[[]]]]
=> 2 = 1 + 1
[1,2,1,3] => [[[[[]]]]]
=> 1 = 0 + 1
[1,2,3,1] => [[[],[[]]]]
=> 1 = 0 + 1
[1,3,1,2] => [[[[[]]]]]
=> 1 = 0 + 1
[1,3,2,1] => [[[[],[]]]]
=> 1 = 0 + 1
[2,1,1,3] => [[],[[[]]]]
=> 2 = 1 + 1
[2,1,3,1] => [[[[[]]]]]
=> 1 = 0 + 1
[2,3,1,1] => [[],[[[]]]]
=> 2 = 1 + 1
[3,1,1,2] => [[[]],[[]]]
=> 2 = 1 + 1
[3,1,2,1] => [[[[[]]]]]
=> 1 = 0 + 1
Description
The number of subtrees.
Matching statistic: St000439
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00056: Parking functions āto Dyck pathā¶ Dyck paths
St000439: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000439: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 2 = 0 + 2
[1,1] => [1,1,0,0]
=> 3 = 1 + 2
[1,2] => [1,0,1,0]
=> 2 = 0 + 2
[2,1] => [1,0,1,0]
=> 2 = 0 + 2
[1,1,1] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,2] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,2,1] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,1] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,3] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[1,3,1] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,1] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[1,2,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,2,1] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[2,1,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[2,3,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1,2] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[3,2,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000800
Mp00056: Parking functions āto Dyck pathā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St000800: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00201: Dyck paths āRingelā¶ Permutations
St000800: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [2,1] => 0
[1,1] => [1,1,0,0]
=> [2,3,1] => 1
[1,2] => [1,0,1,0]
=> [3,1,2] => 0
[2,1] => [1,0,1,0]
=> [3,1,2] => 0
[1,1,1] => [1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,1,2] => [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,2,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[2,1,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,3] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,3,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[3,1,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,2,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[2,1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[2,2,1] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,3,2] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,3,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[3,1,2] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[3,2,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
Description
The number of occurrences of the vincular pattern |231 in a permutation.
This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St001231
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00056: Parking functions āto Dyck pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001231: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001231: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> 0
[1,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[2,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,2] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
Description
The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.
Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.
Matching statistic: St001234
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00056: Parking functions āto Dyck pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001234: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001234: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> 0
[1,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[2,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,2] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
Description
The number of indecomposable three dimensional modules with projective dimension one.
It return zero when there are no such modules.
Matching statistic: St001640
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00056: Parking functions āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St001640: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St001640: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,1,0,0]
=> [1,2] => 1
[1,2] => [1,0,1,0]
=> [2,1] => 0
[2,1] => [1,0,1,0]
=> [2,1] => 0
[1,1,1] => [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,1,2] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,3,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[3,1,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,2,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,2,1] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,3,2] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,1,3] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,3,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,1,2] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,2,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000011
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00056: Parking functions āto Dyck pathā¶ Dyck paths
Mp00101: Dyck paths ādecomposition reverseā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00101: Dyck paths ādecomposition reverseā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
The following 260 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000115The single entry in the last row. St000160The multiplicity of the smallest part of a partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000542The number of left-to-right-minima of a permutation. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001937The size of the center of a parking function. St000738The first entry in the last row of a standard tableau. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000546The number of global descents of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000989The number of final rises of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000068The number of minimal elements in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000383The last part of an integer composition. St000505The biggest entry in the block containing the 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000759The smallest missing part in an integer partition. St000765The number of weak records in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000942The number of critical left to right maxima of the parking functions. St000996The number of exclusive left-to-right maxima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001497The position of the largest weak excedence of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000061The number of nodes on the left branch of a binary tree. St000203The number of external nodes of a binary tree. St000734The last entry in the first row of a standard tableau. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001733The number of weak left to right maxima of a Dyck path. St000444The length of the maximal rise of a Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000502The number of successions of a set partitions. St001948The number of augmented double ascents of a permutation. St000504The cardinality of the first block of a set partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000877The depth of the binary word interpreted as a path. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000717The number of ordinal summands of a poset. St000823The number of unsplittable factors of the set partition. St000990The first ascent of a permutation. St001586The number of odd parts smaller than the largest even part in an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001281The normalized isoperimetric number of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001525The number of symmetric hooks on the diagonal of a partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000259The diameter of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000904The maximal number of repetitions of an integer composition. St000260The radius of a connected graph. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000699The toughness times the least common multiple of 1,. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000177The number of free tiles in the pattern. St000178Number of free entries. St000674The number of hills of a Dyck path. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000225Difference between largest and smallest parts in a partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001383The BG-rank of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001541The Gini index of an integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000441The number of successions of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001651The Frankl number of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001330The hat guessing number of a graph. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000456The monochromatic index of a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001568The smallest positive integer that does not appear twice in the partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000090The variation of a composition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000839The largest opener of a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001487The number of inner corners of a skew partition.
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