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Your data matches 768 different statistics following compositions of up to 3 maps.
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Matching statistic: St000205
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000205: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000205: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,-1] => [2]
=> [1,1]
=> [1]
=> 0
[-2,1] => [2]
=> [1,1]
=> [1]
=> 0
[1,3,-2] => [2]
=> [1,1]
=> [1]
=> 0
[1,-3,2] => [2]
=> [1,1]
=> [1]
=> 0
[-1,3,-2] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,-3,2] => [2,1]
=> [2,1]
=> [1]
=> 0
[2,-1,3] => [2]
=> [1,1]
=> [1]
=> 0
[2,-1,-3] => [2,1]
=> [2,1]
=> [1]
=> 0
[-2,1,3] => [2]
=> [1,1]
=> [1]
=> 0
[-2,1,-3] => [2,1]
=> [2,1]
=> [1]
=> 0
[2,3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[2,-3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-2,-3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[3,-1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-3,-1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,-1] => [2]
=> [1,1]
=> [1]
=> 0
[3,-2,-1] => [2,1]
=> [2,1]
=> [1]
=> 0
[-3,2,1] => [2]
=> [1,1]
=> [1]
=> 0
[-3,-2,1] => [2,1]
=> [2,1]
=> [1]
=> 0
[1,2,4,-3] => [2]
=> [1,1]
=> [1]
=> 0
[1,2,-4,3] => [2]
=> [1,1]
=> [1]
=> 0
[1,-2,4,-3] => [2,1]
=> [2,1]
=> [1]
=> 0
[1,-2,-4,3] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,2,4,-3] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,2,-4,3] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,-2,4,-3] => [2,1,1]
=> [3,1]
=> [1]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 0
[1,3,-2,4] => [2]
=> [1,1]
=> [1]
=> 0
[1,3,-2,-4] => [2,1]
=> [2,1]
=> [1]
=> 0
[1,-3,2,4] => [2]
=> [1,1]
=> [1]
=> 0
[1,-3,2,-4] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,3,-2,4] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,3,-2,-4] => [2,1,1]
=> [3,1]
=> [1]
=> 0
[-1,-3,2,4] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,-3,2,-4] => [2,1,1]
=> [3,1]
=> [1]
=> 0
[1,3,4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,-4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,-4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-1,3,4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[-1,3,-4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[-1,-3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[-1,-3,-4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,-2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,-4,2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,-4,-2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000206: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000206: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,-1] => [2]
=> [1,1]
=> [1]
=> 0
[-2,1] => [2]
=> [1,1]
=> [1]
=> 0
[1,3,-2] => [2]
=> [1,1]
=> [1]
=> 0
[1,-3,2] => [2]
=> [1,1]
=> [1]
=> 0
[-1,3,-2] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,-3,2] => [2,1]
=> [2,1]
=> [1]
=> 0
[2,-1,3] => [2]
=> [1,1]
=> [1]
=> 0
[2,-1,-3] => [2,1]
=> [2,1]
=> [1]
=> 0
[-2,1,3] => [2]
=> [1,1]
=> [1]
=> 0
[-2,1,-3] => [2,1]
=> [2,1]
=> [1]
=> 0
[2,3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[2,-3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-2,-3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[3,-1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-3,-1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,-1] => [2]
=> [1,1]
=> [1]
=> 0
[3,-2,-1] => [2,1]
=> [2,1]
=> [1]
=> 0
[-3,2,1] => [2]
=> [1,1]
=> [1]
=> 0
[-3,-2,1] => [2,1]
=> [2,1]
=> [1]
=> 0
[1,2,4,-3] => [2]
=> [1,1]
=> [1]
=> 0
[1,2,-4,3] => [2]
=> [1,1]
=> [1]
=> 0
[1,-2,4,-3] => [2,1]
=> [2,1]
=> [1]
=> 0
[1,-2,-4,3] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,2,4,-3] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,2,-4,3] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,-2,4,-3] => [2,1,1]
=> [3,1]
=> [1]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 0
[1,3,-2,4] => [2]
=> [1,1]
=> [1]
=> 0
[1,3,-2,-4] => [2,1]
=> [2,1]
=> [1]
=> 0
[1,-3,2,4] => [2]
=> [1,1]
=> [1]
=> 0
[1,-3,2,-4] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,3,-2,4] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,3,-2,-4] => [2,1,1]
=> [3,1]
=> [1]
=> 0
[-1,-3,2,4] => [2,1]
=> [2,1]
=> [1]
=> 0
[-1,-3,2,-4] => [2,1,1]
=> [3,1]
=> [1]
=> 0
[1,3,4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,-4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,-4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[-1,3,4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[-1,3,-4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[-1,-3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[-1,-3,-4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,-2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,-4,2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,-4,-2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000355
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000355: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000355: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Values
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[3,-2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-2,4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-2,-4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-3,2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-3,2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,-4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,-4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-1,3,4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,3,-4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,-4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,4,-2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,-2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-8,-6,-4,-2,1,3,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-5,4,-7,3,8,1,2,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[6,7,4,5,-8,-2,1,3] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-5,4,-6,3,-7,2,-8,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-2,1,-7,3,4,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-2,-7,-6,-5,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[6,-5,2,-7,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,1,-7,-6,2,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 0
[2,3,-7,-6,1,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,1,-6,5,2,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,4,-7,-2,1,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-6,4,-7,-3,-2,1,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000375
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000375: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000375: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Values
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[3,-2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-2,4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-2,-4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-3,2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-3,2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,-4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,-4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-1,3,4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,3,-4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,-4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,4,-2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,-2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-8,-6,-4,-2,1,3,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-5,4,-7,3,8,1,2,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[6,7,4,5,-8,-2,1,3] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-5,4,-6,3,-7,2,-8,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-2,1,-7,3,4,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-2,-7,-6,-5,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[6,-5,2,-7,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,1,-7,-6,2,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 0
[2,3,-7,-6,1,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,1,-6,5,2,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,4,-7,-2,1,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-6,4,-7,-3,-2,1,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000406
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000406: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000406: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Values
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[3,-2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-2,4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-2,-4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-3,2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-3,2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,-4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,-4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-1,3,4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,3,-4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,-4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,4,-2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,-2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-8,-6,-4,-2,1,3,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-5,4,-7,3,8,1,2,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[6,7,4,5,-8,-2,1,3] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-5,4,-6,3,-7,2,-8,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-2,1,-7,3,4,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-2,-7,-6,-5,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[6,-5,2,-7,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,1,-7,-6,2,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 0
[2,3,-7,-6,1,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,1,-6,5,2,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,4,-7,-2,1,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-6,4,-7,-3,-2,1,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
Description
The number of occurrences of the pattern 3241 in a permutation.
Matching statistic: St000407
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000407: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000407: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Values
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[3,-2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-2,4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-2,-4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-3,2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-3,2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,-4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,-4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-1,3,4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,3,-4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,-4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,4,-2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,-2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-8,-6,-4,-2,1,3,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-5,4,-7,3,8,1,2,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[6,7,4,5,-8,-2,1,3] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-5,4,-6,3,-7,2,-8,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-2,1,-7,3,4,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-2,-7,-6,-5,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[6,-5,2,-7,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,1,-7,-6,2,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 0
[2,3,-7,-6,1,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,1,-6,5,2,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,4,-7,-2,1,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-6,4,-7,-3,-2,1,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
Description
The number of occurrences of the pattern 2143 in a permutation.
A permutation $\pi$ avoids this pattern if and only if it is ''vexillary'' as introduced in [1].
Matching statistic: St000425
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000425: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000425: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Values
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[3,-2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-2,4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-2,-4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-3,2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-3,2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,-4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,-4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-1,3,4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,3,-4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,-4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,4,-2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,-2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-8,-6,-4,-2,1,3,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-5,4,-7,3,8,1,2,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[6,7,4,5,-8,-2,1,3] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-5,4,-6,3,-7,2,-8,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-2,1,-7,3,4,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-2,-7,-6,-5,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[6,-5,2,-7,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,1,-7,-6,2,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 0
[2,3,-7,-6,1,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,1,-6,5,2,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,4,-7,-2,1,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-6,4,-7,-3,-2,1,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
Description
The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.
Matching statistic: St000486
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000486: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000486: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Values
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,3,-2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,-3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[3,-2,-1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-2,4,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,-2,-4,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,2,4,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,2,-4,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,-2,4,-3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[-1,-2,-4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2,-4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,-3,2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2,-4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,3,-2,4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,3,-2,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[-1,-3,2,4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,-3,2,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,3,4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,-4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,-4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-1,3,4,-2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[-1,3,-4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[-1,-3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[-1,-3,-4,-2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[1,4,2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,4,-2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,-2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-8,-6,-4,-2,1,3,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-5,4,-7,3,8,1,2,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[6,7,4,5,-8,-2,1,3] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[-5,4,-6,3,-7,2,-8,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1
[-2,1,-7,3,4,5,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-2,-7,-6,-5,1,3,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[6,-5,2,-7,1,3,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[3,1,-7,-6,2,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0
[2,3,-7,-6,1,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1
[-7,1,-6,5,2,3,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[3,4,-7,-2,1,5,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-6,4,-7,-3,-2,1,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St000516
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000516: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000516: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Values
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[3,-2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,4,-3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,2,-4,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-2,4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-2,-4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,-3,2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2,-4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,3,-2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[-1,-3,2,4] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[-1,-3,2,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,-4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,-4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-1,3,4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,3,-4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[-1,-3,-4,-2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,4,-2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,-2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-8,-6,-4,-2,1,3,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-5,4,-7,3,8,1,2,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[6,7,4,5,-8,-2,1,3] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-5,4,-6,3,-7,2,-8,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 0
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-2,1,-7,3,4,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-2,-7,-6,-5,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[6,-5,2,-7,1,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,1,-7,-6,2,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 0
[2,3,-7,-6,1,4,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 1
[-7,1,-6,5,2,3,4] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[3,4,-7,-2,1,5,6] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
[-6,4,-7,-3,-2,1,5] => [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 0
Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St000623
(load all 160 compositions to match this statistic)
(load all 160 compositions to match this statistic)
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000623: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000623: Permutations ā¶ ā¤Result quality: 50% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 50%
Values
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,3,-2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,-3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[3,-2,-1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-2,4,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,-2,-4,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,2,4,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,2,-4,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,-2,4,-3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[-1,-2,-4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,3,-2,-4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,-3,2,4] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,-3,2,-4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,3,-2,4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,3,-2,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[-1,-3,2,4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[-1,-3,2,-4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,3,4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,-4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,4,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-3,-4,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-1,3,4,-2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[-1,3,-4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[-1,-3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[-1,-3,-4,-2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[1,4,2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,4,-2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,-4,-2,-3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-8,-6,-4,-2,1,3,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-5,4,-7,3,8,1,2,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[6,7,4,5,-8,-2,1,3] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[-5,4,-6,3,-7,2,-8,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 1
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1
[-2,1,-7,3,4,5,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-2,-7,-6,-5,1,3,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[6,-5,2,-7,1,3,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[3,1,-7,-6,2,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0
[2,3,-7,-6,1,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1
[-7,1,-6,5,2,3,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[3,4,-7,-2,1,5,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
[-6,4,-7,-3,-2,1,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 0
Description
The number of occurrences of the pattern 52341 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to have a complete parabolic bundle structure [1].
The following 758 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000646The number of big ascents of a permutation. St000650The number of 3-rises of 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. 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. St001466The number of transpositions swapping cyclically adjacent numbers in 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. St000652The maximal difference between successive positions of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a 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. St000570The Edelman-Greene number of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Mƶbius function of a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. St001811The Castelnuovo-Mumford regularity of a permutation. 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 $K[x]/(x^n)$. St001890The maximum magnitude of the Mƶbius function of a poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000091The descent variation of a composition. 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. 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. St001577The minimal number of edges to add or remove to make a graph a cograph. St001596The number of two-by-two squares inside a skew partition. St001793The difference between the clique number and the chromatic number of a graph. St000535The rank-width of a graph. St000775The multiplicity of the largest eigenvalue in a graph. 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. St001597The Frobenius rank of a skew partition. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue 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. 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. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001272The number of graphs with the same degree sequence. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001261The Castelnuovo-Mumford regularity of a graph. St000210Minimum over maximum difference of elements in cycles. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents 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. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001728The number of invisible descents of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St000359The number of occurrences of the pattern 23-1. St000298The order dimension or Dushnik-Miller dimension of a poset. St000002The number of occurrences of the pattern 123 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. 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. St001513The number of nested exceedences of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000054The first entry of the permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000042The number of crossings of a perfect matching. St000296The length of the symmetric border of a binary word. 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. St000326The position of the first one in a binary word after appending a 1 at the end. St000876The number of factors in the Catalan decomposition of a binary word. St000733The row containing the largest entry of a standard tableau. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000485The length of the longest cycle of a permutation. St000352The Elizalde-Pak rank of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000358The number of occurrences of the pattern 31-2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000962The 3-shifted major index of a permutation. St001130The number of two successive successions in 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 $eA$. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000133The "bounce" of a permutation. 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. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000403The Szeged index minus the Wiener index of a graph. St000448The number of pairs of vertices of a graph with distance 2. St000461The rix statistic of a permutation. St000552The number of cut vertices of a graph. St000674The number of hills of a Dyck path. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000974The length of the trunk of an ordered tree. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000989The number of final rises of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. 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. 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001374The Padmakar-Ivan index of a graph. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. 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. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001646The number of edges that can be added without increasing the maximal degree of a graph. 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. St001689The number of celebrities in a graph. 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. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the TurĆ”n graph. St001799The number of proper separations of a graph. St001871The number of triconnected components of a graph. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000093The cardinality of a maximal independent set of vertices of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000544The cop number of a graph. St000553The number of blocks of a graph. St000654The first descent of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000785The number of distinct colouring schemes of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000916The packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St001057The Grundy value of the game of creating an independent set in a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. St001271The competition number of a graph. St001282The number of graphs with the same chromatic polynomial. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001512The minimum rank of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001642The Prague dimension of a graph. St001665The number of pure excedances of a permutation. St001734The lettericity of a graph. St001737The number of descents of type 2 in a permutation. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000062The length of the longest increasing subsequence of the permutation. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St001957The number of Hasse diagrams with a given underlying undirected graph. St000069The number of maximal elements of a poset. St001434The number of negative sum pairs of a signed permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000889The number of alternating sign matrices with the same antidiagonal sums. St000065The number of entries equal to -1 in an alternating sign matrix. St001947The number of ties in a parking function. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001260The permanent of an alternating sign matrix. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St000629The defect of a binary word. St000921The number of internal inversions of a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St000366The number of double descents of a permutation. St000546The number of global descents of a permutation. St000929The constant term of the character polynomial of an integer partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St000058The order of a permutation. St000731The number of double exceedences of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001696The natural major index of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. 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. St000480The number of lower covers of a partition in dominance order. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000995The largest even part of an integer partition. St001092The number of distinct even parts of a partition. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000517The Kreweras number of an integer partition. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St001129The product of the squares of the parts of a partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000009The charge of a standard tableau. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000057The Shynar inversion number of a standard tableau. 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. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000223The number of nestings in the permutation. St000257The number of distinct parts of a partition that occur at least twice. 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 $3$. St000386The number of factors DDU in a Dyck path. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000481The number of upper covers of a partition in dominance order. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000647The number of big descents of a permutation. St000649The number of 3-excedences of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000768The number of peaks in an integer composition. St000779The tier of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000872The number of very big descents of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001394The genus of a permutation. St001430The number of positive entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. 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. St001948The number of augmented double ascents of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000048The multinomial of the parts of a partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000346The number of coarsenings of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000805The number of peaks of the associated bargraph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000886The number of permutations with the same antidiagonal sums. St000996The number of exclusive left-to-right maxima of a permutation. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000842The breadth of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St001555The order of a signed permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. 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. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000496The rcs statistic of a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001172The number of 1-rises at odd height of 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. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001411The number of patterns 321 or 3412 in a permutation. St001584The area statistic between a Dyck path and its bounce path. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. 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. St000047The number of standard immaculate tableaux of a given shape. St000068The number of minimal elements in a poset. St000115The single entry in the last row. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000701The protection number of a binary tree. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000764The number of strong records in an integer composition. St000816The number of standard composition tableaux of the composition. St000905The number of different multiplicities of parts of an integer composition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001732The number of peaks visible from the left. St001761The maximal multiplicity of a letter in a reduced word of a permutation. 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. St001884The number of borders of a binary word. St000397The Strahler number of a rooted tree. St000793The length of the longest partition in the vacillating tableau corresponding to 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. St001141The number of occurrences of hills of size 3 in a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000864The number of circled entries of the shifted recording tableau of a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000488The number of cycles of a permutation of length at most 2. St001081The number of minimal length factorizations of a permutation into star transpositions. 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. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000847The number of standard Young tableaux whose descent set is the binary word. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000999Number of indecomposable projective module with injective 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000630The length of the shortest palindromic decomposition of a binary word. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000016The number of attacking pairs of a standard tableau. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000292The number of ascents of a binary word. St000348The non-inversion sum of a binary word. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000682The Grundy value of Welter's game on a binary word. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001910The height of the middle non-run of a Dyck path. St001961The sum of the greatest common divisors of all pairs of parts. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000026The position of the first return of a Dyck path. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000120The number of left tunnels of a Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000390The number of runs of ones in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000628The balance of a binary word. St000675The number of centered multitunnels of a Dyck path. St000691The number of changes of a binary word. St000734The last entry in the first row of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by 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. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001462The number of factors of a standard tableaux under concatenation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001595The number of standard Young tableaux of the skew partition. St001721The degree of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000015The number of peaks of a Dyck path. St000439The position of the first down step of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001570The minimal number of edges to add to make a graph Hamiltonian. 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 $eA$. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000369The dinv deficit of a Dyck path. St000567The sum of the products of all pairs of parts. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000444The length of the maximal rise of a Dyck path. 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. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St001485The modular major index of a binary word. St000183The side length of the Durfee square of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001889The size of the connectivity set of a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001851The number of Hecke atoms of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001768The number of reduced words of a signed permutation. St001624The breadth of a lattice. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001964The interval resolution global dimension of a poset. 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. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000911The number of maximal antichains of maximal size in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000717The number of ordinal summands of a poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000635The number of strictly order preserving maps of a poset into itself. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000084The number of subtrees. St000328The maximum number of child nodes in a tree.
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