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Your data matches 353 different statistics following compositions of up to 3 maps.
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Matching statistic: St000377
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000377: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000377: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000681
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000681: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000681: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000937
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000937: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000937: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St001091
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001091: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001091: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St001176
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001176: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001176: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
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: St001382
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001382: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001382: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Matching statistic: St001384
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001384: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001384: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St001714
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001714: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St001714: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 2
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 2
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 2
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 3
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 2
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 3 = 2 + 1
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000148
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(load all 2 compositions to match this statistic)
Mp00260: Signed permutations āDemazure product with inverseā¶ Signed permutations
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000148: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00169: Signed permutations āodd cycle typeā¶ Integer partitions
St000148: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 3 = 2 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
=> 3 = 2 + 1
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
=> 4 = 3 + 1
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
=> 3 = 2 + 1
Description
The number of odd parts of a partition.
The following 343 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000475The number of parts equal to 1 in a partition. St000531The leading coefficient of the rook polynomial of an integer 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000993The multiplicity of the largest part of an integer partition. St001127The sum of the squares of the parts of a 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. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001933The largest multiplicity of a part in an integer partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St001400The total number of Littlewood-Richardson tableaux of given shape. St000012The area of a Dyck path. St000015The number of peaks of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000144The pyramid weight of the Dyck path. St000145The Dyson rank of a partition. St000157The number of descents of a standard tableau. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000876The number of factors in the Catalan decomposition of a binary word. 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. St000921The number of internal inversions of a binary word. St000984The number of boxes below precisely one peak. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000013The height of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000392The length of the longest run of ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000474Dyson's crank of a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000477The weight of a partition according to Alladi. St000519The largest length of a factor maximising the subword complexity. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000770The major index of an integer partition when read from bottom to top. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000922The minimal number such that all substrings of this length are unique. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St000979Half of MacMahon's equal index of a Dyck path. St000982The length of the longest constant subword. St000992The alternating sum of the parts of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. 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. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001279The sum of the parts of an integer partition that are at least two. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. 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. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001959The product of the heights of the peaks of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000874The position of the last double rise in a Dyck path. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. 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. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension 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. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. 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. St001658The total number of rook placements on a Ferrers board. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001814The number of partitions interlacing the given partition. St000438The position of the last up step in a Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra 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. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001838The number of nonempty primitive factors of a binary word. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001896The number of right descents of a signed permutations. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001429The number of negative entries in a signed permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001427The number of descents of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000735The last entry on the main diagonal of a standard tableau. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. 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. St000264The girth of a graph, which is not a tree. St000478Another weight of a partition according to Alladi. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. 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. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given 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. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001819The flag Denert index of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000080The rank of the poset. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001060The distinguishing index of a graph. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000259The diameter of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000456The monochromatic index of a connected graph. St000302The determinant of the distance matrix of a connected graph. St001545The second Elser number of a connected graph. St000467The hyper-Wiener index of a connected graph. St000464The Schultz index of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph.
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