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Your data matches 643 different statistics following compositions of up to 3 maps.
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Matching statistic: St000807
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000807: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000807: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 0
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 0
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 0
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 0
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 0
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 0
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 0
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 0
Description
The sum of the heights of the valleys of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St000805
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000805: Integer compositions ā¶ ā¤Result quality: 95% āvalues known / values provided: 95%ādistinct values known / distinct values provided: 100%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
St000805: Integer compositions ā¶ ā¤Result quality: 95% āvalues known / values provided: 95%ādistinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 0 + 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1 = 0 + 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 0 + 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1 = 0 + 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1 = 0 + 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1 = 0 + 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 0 + 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1 = 0 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => 1 = 0 + 1
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1 = 0 + 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 0 + 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1 = 0 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 0 + 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1 = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => 1 = 0 + 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1 = 0 + 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 0 + 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1 = 0 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 0 + 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 1 = 0 + 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1 = 0 + 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1 = 0 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1 = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[5,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[4,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[4,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[6,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[5,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[5,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[3,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[7,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[6,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[6,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[5,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[5,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[5,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[4,4,1,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[4,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[4,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[4,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[4,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[3,3,3,1,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,3,2,2,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,3,2,1,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[3,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[3,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[2,2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
[2,2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? = 0 + 1
[8,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ? = 0 + 1
Description
The number of peaks of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.
Matching statistic: St000687
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00313: Integer partitions āGlaisher-Franklin inverseā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000687: Dyck paths ā¶ ā¤Result quality: 49% āvalues known / values provided: 49%ādistinct values known / distinct values provided: 50%
Mp00313: Integer partitions āGlaisher-Franklin inverseā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000687: Dyck paths ā¶ ā¤Result quality: 49% āvalues known / values provided: 49%ādistinct values known / distinct values provided: 50%
Values
[1,1]
=> [1]
=> [1]
=> [1,0]
=> 0
[2,1]
=> [1]
=> [1]
=> [1,0]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[3,1]
=> [1]
=> [1]
=> [1,0]
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[4,1]
=> [1]
=> [1]
=> [1,0]
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[5,1]
=> [1]
=> [1]
=> [1,0]
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[6,1]
=> [1]
=> [1]
=> [1,0]
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[4,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[7,1]
=> [1]
=> [1]
=> [1,0]
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 0
[6,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[5,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,3,1]
=> [3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3,3,2]
=> [3,3,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[4,4,4]
=> [4,4]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,4,3,1]
=> [4,3,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,4,2,2]
=> [4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,4,2,1,1]
=> [4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,3,3,2]
=> [3,3,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0
[5,5,2,1]
=> [5,2,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
[5,4,4]
=> [4,4]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,4,3,1]
=> [4,3,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,4,2,2]
=> [4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,4,2,1,1]
=> [4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,3,3,2]
=> [3,3,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[4,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
[4,4,3,2]
=> [4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,4,3,1,1]
=> [4,3,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,4,2,2,1]
=> [4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,3,3,2,1]
=> [3,3,2,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 0
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[3,3,3,3,1]
=> [3,3,3,1]
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 0
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 0
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0
[3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0
[6,5,2,1]
=> [5,2,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
[6,4,4]
=> [4,4]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,4,3,1]
=> [4,3,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,4,2,2]
=> [4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,4,2,1,1]
=> [4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,3,3,2]
=> [3,3,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[5,5,4]
=> [5,4]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,5,2,1,1]
=> [5,2,1,1]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 0
[5,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
[5,4,3,2]
=> [4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,4,3,1,1]
=> [4,3,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,4,2,2,1]
=> [4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.
In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules.
This statistic was discussed in [Theorem 5.7, 1].
Matching statistic: St001107
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00313: Integer partitions āGlaisher-Franklin inverseā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001107: Dyck paths ā¶ ā¤Result quality: 33% āvalues known / values provided: 33%ādistinct values known / distinct values provided: 50%
Mp00313: Integer partitions āGlaisher-Franklin inverseā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001107: Dyck paths ā¶ ā¤Result quality: 33% āvalues known / values provided: 33%ādistinct values known / distinct values provided: 50%
Values
[1,1]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1]
=> [2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1]
=> [3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[2,1,1]
=> [3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[3,2]
=> [2,2,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[3,1,1]
=> [3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,2,1]
=> [3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,1,1,1]
=> [4,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1]
=> [5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[4,2]
=> [2,2,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[4,1,1]
=> [3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,3]
=> [2,2,2]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[3,1,1,1]
=> [4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
[2,2,2]
=> [3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 0
[2,2,1,1]
=> [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[2,1,1,1,1]
=> [5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [6]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 0
[5,2]
=> [2,2,1,1,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[4,3]
=> [2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[4,2,1]
=> [3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[4,1,1,1]
=> [4,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[3,3,1]
=> [3,2,2]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[3,2,2]
=> [3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[3,1,1,1,1]
=> [5,1,1]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 0
[2,2,2,1]
=> [4,3]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 0
[2,2,1,1,1]
=> [5,2]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 0
[2,1,1,1,1,1]
=> [6,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[4,4]
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
[4,3,1]
=> [3,2,2,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[4,2,2]
=> [3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 0
[3,3,2]
=> [3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 0
[3,3,1,1]
=> [4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0
[3,2,2,1]
=> [4,3,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 0
[2,2,2,2]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[2,1,1,1,1,1,1]
=> [7,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,1,1,1,1]
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,2,1,1]
=> [4,2,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[3,2,2,2]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[3,1,1,1,1,1,1]
=> [7,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0
[2,2,1,1,1,1,1]
=> [7,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1]
=> [9]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[6,2,1,1]
=> [4,2,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,3,2,1]
=> [4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0
[4,2,2,2]
=> [4,4,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 0
[4,1,1,1,1,1,1]
=> [7,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 0
[3,3,2,2]
=> [4,4,2]
=> [8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[3,2,1,1,1,1,1]
=> [7,2,1]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 0
[2,2,2,2,2]
=> [5,5]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[2,2,2,1,1,1,1]
=> [7,3]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,4,1,1]
=> [4,2,2,2,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 0
[5,3,2,1]
=> [4,3,2,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
[5,2,2,2]
=> [4,4,1,1,1]
=> [8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 0
[5,1,1,1,1,1,1]
=> [7,1,1,1,1]
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0
[4,3,2,2]
=> [4,4,2,1]
=> [8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0
[4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[3,3,3,2]
=> [4,4,3]
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 1
[3,3,1,1,1,1,1]
=> [7,2,2]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 0
[3,2,2,2,2]
=> [5,5,1]
=> [10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[3,2,2,1,1,1,1]
=> [7,3,1]
=> [7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[2,2,2,2,1,1,1]
=> [7,4]
=> [7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [8,3]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[8,2,1,1]
=> [4,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[7,2,2,1]
=> [4,3,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
[6,4,1,1]
=> [4,2,2,2,1,1]
=> [4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 0
[6,3,2,1]
=> [4,3,2,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[6,2,2,2]
=> [4,4,1,1,1,1]
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[6,1,1,1,1,1,1]
=> [7,1,1,1,1,1]
=> [7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 0
[5,3,2,2]
=> [4,4,2,1,1]
=> [8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 0
[5,2,1,1,1,1,1]
=> [7,2,1,1,1]
=> [7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0
[5,1,1,1,1,1,1,1]
=> [8,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000232
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000232: Set partitions ā¶ ā¤Result quality: 31% āvalues known / values provided: 31%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000232: Set partitions ā¶ ā¤Result quality: 31% āvalues known / values provided: 31%ādistinct values known / distinct values provided: 50%
Values
[1,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[3,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[5,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[4,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[6,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[5,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[4,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[6,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[5,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[4,4,3,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 0
[4,4,2,2]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
[4,4,2,1,1]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 0
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
[4,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[4,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[4,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? = 0
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> {{1,2,3},{4,5},{6},{7},{8},{9}}
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> {{1,2},{3,4},{5,6},{7,8},{9},{10}}
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> {{1,2},{3,4},{5,6},{7},{8},{9},{10}}
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000233
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000233: Set partitions ā¶ ā¤Result quality: 31% āvalues known / values provided: 31%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000233: Set partitions ā¶ ā¤Result quality: 31% āvalues known / values provided: 31%ādistinct values known / distinct values provided: 50%
Values
[1,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[3,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[5,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[4,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[6,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[5,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[4,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[6,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[5,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[4,4,3,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 0
[4,4,2,2]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
[4,4,2,1,1]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 0
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
[4,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[4,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[4,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? = 0
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> {{1,2,3},{4,5},{6},{7},{8},{9}}
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> {{1,2},{3,4},{5,6},{7,8},{9},{10}}
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> {{1,2},{3,4},{5,6},{7},{8},{9},{10}}
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000496
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000496: Set partitions ā¶ ā¤Result quality: 31% āvalues known / values provided: 31%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000496: Set partitions ā¶ ā¤Result quality: 31% āvalues known / values provided: 31%ādistinct values known / distinct values provided: 50%
Values
[1,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[3,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[5,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[4,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[6,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[5,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[4,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[6,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[5,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[4,4,3,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 0
[4,4,2,2]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
[4,4,2,1,1]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 0
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
[4,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[4,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[4,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? = 0
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> {{1,2,3},{4,5},{6},{7},{8},{9}}
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> {{1,2},{3,4},{5,6},{7,8},{9},{10}}
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> {{1,2},{3,4},{5,6},{7},{8},{9},{10}}
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
Description
The rcs statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St000091
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
St000091: Integer compositions ā¶ ā¤Result quality: 29% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
St000091: Integer compositions ā¶ ā¤Result quality: 29% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 50%
Values
[1,1]
=> [1]
=> [[1]]
=> [1] => 0
[2,1]
=> [1]
=> [[1]]
=> [1] => 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0
[3,1]
=> [1]
=> [[1]]
=> [1] => 0
[2,2]
=> [2]
=> [[1,2]]
=> [2] => 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0
[4,1]
=> [1]
=> [[1]]
=> [1] => 0
[3,2]
=> [2]
=> [[1,2]]
=> [2] => 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0
[5,1]
=> [1]
=> [[1]]
=> [1] => 0
[4,2]
=> [2]
=> [[1,2]]
=> [2] => 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [3] => 0
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 0
[6,1]
=> [1]
=> [[1]]
=> [1] => 0
[5,2]
=> [2]
=> [[1,2]]
=> [2] => 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [3] => 0
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 0
[7,1]
=> [1]
=> [[1]]
=> [1] => 0
[6,2]
=> [2]
=> [[1,2]]
=> [2] => 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [3] => 0
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ? = 0
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ? = 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,1,1] => ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [2,2,1,1,1,1] => ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => ? = 0
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ? = 0
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 0
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 0
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 0
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ? = 0
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,1,1] => ? = 0
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [2,2,1,1,1,1] => ? = 0
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 0
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ? = 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [2,2,2,2,1] => ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [2,2,2,1,1,1] => ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [2,2,1,1,1,1,1] => ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [2,1,1,1,1,1,1,1] => ? = 0
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => ? = 0
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ? = 0
[4,4,3,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ? = 0
[4,4,2,2]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 0
[4,4,2,1,1]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 0
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 0
[4,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ? = 1
[4,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ? = 0
[4,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 0
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 0
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 0
[4,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ? = 0
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,1,1] => ? = 0
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [2,2,1,1,1,1] => ? = 0
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 0
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [3,3,3] => ? = 0
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [3,3,1,1,1] => ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [3,2,2,2] => ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [3,2,2,1,1] => ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [3,2,1,1,1,1] => ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [3,1,1,1,1,1,1] => ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [2,2,2,2,1] => ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [2,2,2,1,1,1] => ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [2,2,1,1,1,1,1] => ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [2,1,1,1,1,1,1,1] => ? = 0
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => ? = 0
Description
The descent variation of a composition.
Defined in [1].
Matching statistic: St001781
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St001781: Set partitions ā¶ ā¤Result quality: 29% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St001781: Set partitions ā¶ ā¤Result quality: 29% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 50%
Values
[1,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[3,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[5,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[4,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[6,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[5,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[4,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[6,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[5,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[4,4,3,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 0
[4,4,2,2]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
[4,4,2,1,1]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 0
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
[4,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[4,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[4,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? = 0
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> {{1,2,3},{4,5},{6},{7},{8},{9}}
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
Description
The interlacing number of a set partition.
Let $\pi$ be a set partition of $\{1,\dots,n\}$ with $k$ blocks. To each block of $\pi$ we add the element $\infty$, which is larger than $n$. Then, an interlacing of $\pi$ is a pair of blocks $B=(B_1 < \dots < B_b < B_{b+1} = \infty)$ and $C=(C_1 < \dots < C_c < C_{c+1} = \infty)$ together with an index $1\leq i\leq \min(b, c)$, such that $B_i < C_i < B_{i+1} < C_{i+1}$.
Matching statistic: St001839
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St001839: Set partitions ā¶ ā¤Result quality: 29% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St001839: Set partitions ā¶ ā¤Result quality: 29% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 50%
Values
[1,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[2,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[3,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[5,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[4,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[6,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[5,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[4,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7,1]
=> [1]
=> [[1]]
=> {{1}}
=> 0
[6,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[5,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[4,4,3,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 0
[4,4,2,2]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
[4,4,2,1,1]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 0
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
[4,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[4,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 0
[4,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 0
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 0
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 0
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 0
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? = 0
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> {{1,2,3},{4,5},{6},{7},{8},{9}}
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
Description
The number of excedances of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1 \dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
Let $\bar w$ be the nondecreasing rearrangement of $w$.
The word $w$ has an excedance at position $i$ if $w_i > \bar w_i$.
The following 633 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000296The length of the symmetric border of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000355The number of occurrences of the pattern 21-3. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St000317The cycle descent number of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001705The number of occurrences of the pattern 2413 in a permutation. St000908The length of the shortest maximal antichain in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000297The number of leading ones in a binary word. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001396Number of triples of incomparable elements in a finite poset. St000914The sum of the values of the Mƶbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000570The Edelman-Greene number of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000382The first part of an integer composition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000052The number of valleys of a Dyck path not on the x-axis. St001811The Castelnuovo-Mumford regularity of a permutation. St001890The maximum magnitude of the Mƶbius function of a poset. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000842The breadth of a permutation. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001513The number of nested exceedences of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001964The interval resolution global dimension of a poset. St000358The number of occurrences of the pattern 31-2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000701The protection number of a binary tree. St000011The number of touch points (or returns) of a Dyck path. St001730The number of times the path corresponding to a binary word crosses the base line. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001696The natural major index of a standard Young tableau. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000042The number of crossings of a perfect matching. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St000733The row containing the largest entry of a standard tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000478Another weight of a partition according to Alladi. St000629The defect of a binary word. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001732The number of peaks visible from the left. St000929The constant term of the character polynomial of an integer partition. St001175The size of a partition minus the hook length of the base cell. St000480The number of lower covers of a partition in dominance order. St000993The multiplicity of the largest part of an integer partition. St001280The number of parts of an integer partition that are at least two. St000068The number of minimal elements in a poset. St000057The Shynar inversion number of a standard tableau. St000218The number of occurrences of the pattern 213 in a permutation. St000534The number of 2-rises of a permutation. St000731The number of double exceedences of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000871The number of very big ascents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000183The side length of the Durfee square of an integer partition. St000913The number of ways to refine the partition into singletons. St001092The number of distinct even parts of a partition. St001587Half of the largest even part of an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000147The largest part of an integer partition. St000668The least common multiple of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000995The largest even part of an integer partition. St000115The single entry in the last row. St000742The number of big ascents of a permutation after prepending zero. St000651The maximal size of a rise in a permutation. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000781The number of proper colouring schemes of a Ferrers diagram. St000847The number of standard Young tableaux whose descent set is the binary word. St000883The number of longest increasing subsequences of a permutation. St000983The length of the longest alternating subword. St001313The number of Dyck paths above the lattice path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001838The number of nonempty primitive factors of a binary word. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000674The number of hills of a Dyck path. St000879The number of long braid edges in the graph of braid moves of a permutation. 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. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001204Call 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001381The fertility of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000056The decomposition (or block) number of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000542The number of left-to-right-minima of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St000386The number of factors DDU in a Dyck path. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000657The smallest part of an integer composition. St001394The genus of a permutation. St000054The first entry of the permutation. St000374The number of exclusive right-to-left minima of a permutation. St000402Half the size of the symmetry class of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000769The major index of a composition regarded as a word. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000768The number of peaks in an integer composition. St000764The number of strong records in an integer composition. St000766The number of inversions of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000359The number of occurrences of the pattern 23-1. St000761The number of ascents in an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000762The sum of the positions of the weak records of an integer composition. St000765The number of weak records in an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001729The number of visible descents of a permutation. St001665The number of pure excedances of a permutation. St000022The number of fixed points of a permutation. St000214The number of adjacencies of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000383The last part of an integer composition. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000862The number of parts of the shifted shape of a permutation. St001260The permanent of an alternating sign matrix. St000647The number of big descents of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000451The length of the longest pattern of the form k 1 2. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000754The Grundy value for the game of removing nestings in a perfect matching. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000264The girth of a graph, which is not a tree. St000002The number of occurrences of the pattern 123 in a permutation. St000787The number of flips required to make a perfect matching noncrossing. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. 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. St000529The number of permutations whose descent word is the given binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001231The 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St000335The difference of lower and upper interactions. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000058The order of a permutation. St000098The chromatic number of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000640The rank of the largest boolean interval in a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000237The number of small exceedances. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000225Difference between largest and smallest parts in a partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001075The minimal size of a block of a set partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000069The number of maximal elements of a poset. St000028The number of stack-sorts needed to sort a permutation. St000251The number of nonsingleton blocks of a set partition. St000919The number of maximal left branches of a binary tree. St001130The number of two successive successions in a permutation. St000035The number of left outer peaks of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000051The size of the left subtree of a binary tree. St000065The number of entries equal to -1 in an alternating sign matrix. St000090The variation of a composition. St000095The number of triangles of a graph. St000117The number of centered tunnels of a Dyck path. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000124The cardinality of the preimage of the Simion-Schmidt map. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000133The "bounce" of a permutation. St000210Minimum over maximum difference of elements in cycles. St000241The number of cyclical small excedances. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000295The length of the border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000369The dinv deficit of a Dyck path. St000370The genus of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000552The number of cut vertices of a graph. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000948The chromatic discriminant of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001071The beta invariant of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001429The number of negative entries in a signed permutation. St001520The number of strict 3-descents. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000096The number of spanning trees of a graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000363The number of minimal vertex covers of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000487The length of the shortest cycle of a permutation. St000535The rank-width of a graph. St000627The exponent of a binary word. St000654The first descent of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000740The last entry of a permutation. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000843The decomposition number of a perfect matching. St000916The packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000946The sum of the skew hook positions in a Dyck path. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001081The number of minimal length factorizations of a permutation into star transpositions. 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. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001272The number of graphs with the same degree sequence. St001273The projective dimension of the first term in an injective coresolution of the regular module. 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. St001316The domatic number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001597The Frobenius rank of a skew partition. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001694The number of maximal dissociation sets in a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001828The Euler characteristic of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000062The length of the longest increasing subsequence of the permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000636The hull number of a graph. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000918The 2-limited packing number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001029The size of the core of a graph. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001062The maximal size of a block of a set partition. St001109The number of proper colourings of a graph with as few colours as possible. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001654The monophonic hull number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000353The number of inner valleys of a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000962The 3-shifted major index of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001737The number of descents of type 2 in a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St001323The independence gap of a graph. St001282The number of graphs with the same chromatic polynomial. St001621The number of atoms of a lattice. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000963The 2-shifted major index of a permutation. St000886The number of permutations with the same antidiagonal sums. St001354The number of series nodes in the modular decomposition of a graph. St000501The size of the first part in the decomposition of a permutation. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000260The radius of a connected graph. St000717The number of ordinal summands of a poset. St000322The skewness of a graph. St000961The shifted major index of a permutation. St001114The number of odd descents of a permutation. St001536The number of cyclic misalignments of a permutation. St001947The number of ties in a parking function. St001434The number of negative sum pairs of a signed permutation. St000526The number of posets with combinatorially isomorphic order polytopes. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001557The number of inversions of the second entry of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001948The number of augmented double ascents of a permutation. St001957The number of Hasse diagrams with a given underlying undirected graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000652The maximal difference between successive positions of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001335The cardinality of a minimal cycle-isolating set of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001042The size of the automorphism group of the leaf labelled binary unordered tree associated with the perfect matching. St000454The largest eigenvalue of a graph if it is integral. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000635The number of strictly order preserving maps of a poset into itself. St000039The number of crossings of a permutation. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001783The number of odd automorphisms of a graph. St000092The number of outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. 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. St001518The number of graphs with the same ordinary spectrum as the given graph. St000401The size of the symmetry class of a permutation. St000638The number of up-down runs of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000219The number of occurrences of the pattern 231 in a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph.
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