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Your data matches 335 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00204: Permutations āLLPSā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000147
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00044: Integer partitions āconjugateā¶ Integer partitions
St000147: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00044: Integer partitions āconjugateā¶ Integer partitions
St000147: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> [1,1]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [2]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> [1,1,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [2,1]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [2,1]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [3]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [4]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [4,1]
=> 4
Description
The largest part of an integer partition.
Matching statistic: St000378
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00322: Integer partitions āLoehr-Warringtonā¶ Integer partitions
St000378: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00322: Integer partitions āLoehr-Warringtonā¶ Integer partitions
St000378: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> [1,1]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [2]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> [1,1,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [3]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [3]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [3]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [2,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [2,2]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [2,2]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [3,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [3,1,1]
=> 4
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St001176
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
St001176: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Mp00204: Permutations āLLPSā¶ Integer partitions
St001176: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3 = 4 - 1
[]
=> [] => []
=> ? = 0 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000063
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000063: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000063: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> []
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> []
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> []
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> []
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 4
[]
=> [] => []
=> ?
=> ? = 0
Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St000108
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000108: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000108: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> []
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> []
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> []
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> []
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 4
[]
=> [] => []
=> ?
=> ? = 0
Description
The number of partitions contained in the given partition.
Matching statistic: St000532
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000532: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000532: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> []
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> []
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> []
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> []
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 4
[]
=> [] => []
=> ?
=> ? = 0
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St001389
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00044: Integer partitions āconjugateā¶ Integer partitions
St001389: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00044: Integer partitions āconjugateā¶ Integer partitions
St001389: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> [1,1]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [2]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> [1,1,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [2,1]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [2,1]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [3]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [4]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [4,1]
=> 4
[]
=> [] => []
=> []
=> ? = 0
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001400
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St001400: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St001400: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> []
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> []
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> []
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> []
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 4
[]
=> [] => []
=> ?
=> ? = 0
Description
The total number of Littlewood-Richardson tableaux of given shape.
This is the multiplicity of the Schur function $s_\lambda$ in $\sum_{\mu, \nu} s_\mu s_\nu$, where the sum is over all partitions $\mu$ and $\nu$.
Matching statistic: St000148
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000148: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Mp00204: Permutations āLLPSā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000148: Integer partitions ā¶ ā¤Result quality: 92% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2]
=> []
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[]
=> [] => []
=> ?
=> ? = 0 - 1
Description
The number of odd parts of a partition.
The following 325 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001933The largest multiplicity of a part in an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000519The largest length of a factor maximising the subword complexity. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000377The dinv defect of an integer partition. St000288The number of ones in a binary word. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000507The number of ascents of a standard tableau. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000691The number of changes of a binary word. St000097The order of the largest clique of the graph. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000012The area of a Dyck path. St000984The number of boxes below precisely one peak. St000013The height of a Dyck path. St000676The number of odd rises of a Dyck path. St000157The number of descents of a standard tableau. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000098The chromatic number of a graph. St000011The number of touch points (or returns) of a Dyck path. St001581The achromatic number of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001733The number of weak left to right maxima of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000439The position of the first down step of a Dyck path. St001622The number of join-irreducible elements of a lattice. St000306The bounce count of a Dyck path. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001651The Frankl number of a lattice. St000167The number of leaves of an ordered tree. St000662The staircase size of the code of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001875The number of simple modules with projective dimension at most 1. St000502The number of successions of a set partitions. St000925The number of topologically connected components of a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000245The number of ascents of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000211The rank of the set partition. St000105The number of blocks in the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000808The number of up steps of the associated bargraph. 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:
St001494The Alon-Tarsi number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000234The number of global ascents of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001479The number of bridges of a graph. St000728The dimension of a set partition. St000069The number of maximal elements of a poset. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000071The number of maximal chains in a poset. St000527The width of the poset. St000996The number of exclusive left-to-right maxima of a permutation. St000007The number of saliances of the permutation. St000159The number of distinct parts of the integer partition. St000068The number of minimal elements in a poset. St000632The jump number of the poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000015The number of peaks of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001812The biclique partition number of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000495The number of inversions of distance at most 2 of a permutation. St000246The number of non-inversions of a permutation. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001461The number of topologically connected components of the chord diagram of a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000539The number of odd inversions of a permutation. St000546The number of global descents of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000654The first descent of a permutation. St000120The number of left tunnels of a Dyck path. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000214The number of adjacencies of a permutation. St000153The number of adjacent cycles of a permutation. St000203The number of external nodes of a binary tree. St000740The last entry of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001397Number of pairs of incomparable elements in a finite poset. St000081The number of edges of a graph. St000702The number of weak deficiencies of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000237The number of small exceedances. St000308The height of the tree associated to a permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000164The number of short pairs. St001427The number of descents of a signed permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000653The last descent of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000542The number of left-to-right-minima of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000636The hull number of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001654The monophonic hull number of a graph. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000201The number of leaf nodes in a binary tree. St000389The number of runs of ones of odd length in a binary word. St000742The number of big ascents of a permutation after prepending zero. St000822The Hadwiger number of the graph. St000990The first ascent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000004The major index of a permutation. St000080The rank of the poset. St000224The sorting index of a permutation. St000331The number of upper interactions of a Dyck path. St000386The number of factors DDU in a Dyck path. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. 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 $eA$. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001315The dissociation number of a graph. St001428The number of B-inversions of a signed permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001869The maximum cut size of a graph. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. St000061The number of nodes on the left branch of a binary tree. St001346The number of parking functions that give the same permutation. St001674The number of vertices of the largest induced star graph in the 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. St001323The independence gap of a graph. St000619The number of cyclic descents of a permutation. St000731The number of double exceedences of a permutation. St000451The length of the longest pattern of the form k 1 2. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000710The number of big deficiencies of a permutation. St001330The hat guessing number of a graph. St000711The number of big exceedences of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000552The number of cut vertices of a graph. St001692The number of vertices with higher degree than the average degree in a graph. St000236The number of cyclical small weak excedances. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000039The number of crossings of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001152The number of pairs with even minimum in a perfect matching. St001429The number of negative entries in a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001668The number of points of the poset minus the width of the poset. St001948The number of augmented double ascents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001820The size of the image of the pop stack sorting operator. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000445The number of rises of length 1 of a Dyck path. St000647The number of big descents of a permutation. St000884The number of isolated descents of a permutation. St001115The number of even descents of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000242The number of indices that are not cyclical small weak excedances. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001649The length of a longest trail in a graph. St000144The pyramid weight of the Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000942The number of critical left to right maxima of the parking functions. 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. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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