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Your data matches 121 different statistics following compositions of up to 3 maps.
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Matching statistic: St000378
Mp00275: Graphs āto edge-partition of connected componentsā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000378: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000378: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [2]
=> []
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> 0
([(1,3),(2,3)],4)
=> [2]
=> []
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> []
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> []
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> 0
([(2,4),(3,4)],5)
=> [2]
=> []
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> []
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> []
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> []
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> []
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> []
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> []
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> []
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> 0
([(3,5),(4,5)],6)
=> [2]
=> []
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> []
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> []
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> []
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> []
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> []
=> 0
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000442
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00275: Graphs āto edge-partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000442: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000442: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,0,1,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000013
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00275: Graphs āto edge-partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000013: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000013: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000444
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00275: Graphs āto edge-partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000444: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000444: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000052
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St000052: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St000052: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000306
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs āto edge-partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St000306: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St000306: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000373
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
St000373: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
St000373: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000682
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00134: Standard tableaux ādescent wordā¶ Binary words
St000682: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00134: Standard tableaux ādescent wordā¶ Binary words
St000682: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 00 => 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 00 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 000 => 0
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 000 => 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 000 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 000 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 000 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 000 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0100 => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0100 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 11100 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 10000 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> [[1,2,3,4,5,6]]
=> 00000 => 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 11010 => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 11100 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 10000 => 0
Description
The Grundy value of Welter's game on a binary word.
Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.
Matching statistic: St000711
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St000711: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St000711: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St000874
Mp00275: Graphs āto edge-partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00028: Dyck paths āreverseā¶ Dyck paths
St000874: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00028: Dyck paths āreverseā¶ Dyck paths
St000874: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
Description
The position of the last double rise in a Dyck path.
If the Dyck path has no double rises, this statistic is $0$.
The following 111 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000931The number of occurrences of the pattern UUU in a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St000011The number of touch points (or returns) of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St000691The number of changes of a binary word. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000025The number of initial rises of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001733The number of weak left to right maxima of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000734The last entry in the first row of a standard tableau. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000379The number of Hamiltonian cycles in a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000331The number of upper interactions of a Dyck path. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St001271The competition number of a graph. St000668The least common multiple of the parts of the partition. 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. 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. St001175The size of a partition minus the hook length of the base cell. St001248Sum of the even parts of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000464The Schultz index of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001845The number of join irreducibles minus the rank of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001520The number of strict 3-descents. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St000699The toughness times the least common multiple of 1,. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St000934The 2-degree of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition.
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