searching the database
Your data matches 377 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000162
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
Mp00080: Set partitions āto permutationā¶ Permutations
St000162: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000162: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = -1 + 1
{{1,2}}
=> [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => 0 = -1 + 1
{{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => 0 = -1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = -1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => 1 = 0 + 1
Description
The number of nontrivial cycles in the cycle decomposition of a permutation.
This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St000352
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00080: Set partitions āto permutationā¶ Permutations
St000352: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000352: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = -1 + 1
{{1,2}}
=> [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => 0 = -1 + 1
{{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => 0 = -1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = -1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => 1 = 0 + 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St000374
(load all 100 compositions to match this statistic)
(load all 100 compositions to match this statistic)
Mp00080: Set partitions āto permutationā¶ Permutations
St000374: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000374: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = -1 + 1
{{1,2}}
=> [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => 0 = -1 + 1
{{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => 0 = -1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = -1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => 1 = 0 + 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000480
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00079: Set partitions āshapeā¶ Integer partitions
St000480: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000480: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> 0 = -1 + 1
{{1,2}}
=> [2]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> 0 = -1 + 1
{{1,2,3}}
=> [3]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2}}
=> [2,1]
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,1,1]
=> 0 = -1 + 1
{{1,2,3,4}}
=> [4]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1]
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1]
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0 = -1 + 1
{{1,2,3,4,5}}
=> [5]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [6]
=> 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [4,1,1]
=> 1 = 0 + 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000703
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00080: Set partitions āto permutationā¶ Permutations
St000703: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000703: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = -1 + 1
{{1,2}}
=> [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => 0 = -1 + 1
{{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => 0 = -1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = -1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => 1 = 0 + 1
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000994
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00080: Set partitions āto permutationā¶ Permutations
St000994: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000994: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = -1 + 1
{{1,2}}
=> [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => 0 = -1 + 1
{{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => 0 = -1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = -1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => 1 = 0 + 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St001269
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00080: Set partitions āto permutationā¶ Permutations
St001269: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001269: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = -1 + 1
{{1,2}}
=> [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => 0 = -1 + 1
{{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => 0 = -1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = -1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => 1 = 0 + 1
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St001280
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00079: Set partitions āshapeā¶ Integer partitions
St001280: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001280: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> 0 = -1 + 1
{{1,2}}
=> [2]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> 0 = -1 + 1
{{1,2,3}}
=> [3]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2}}
=> [2,1]
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,1,1]
=> 0 = -1 + 1
{{1,2,3,4}}
=> [4]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1]
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1]
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0 = -1 + 1
{{1,2,3,4,5}}
=> [5]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [6]
=> 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [4,1,1]
=> 1 = 0 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001665
(load all 84 compositions to match this statistic)
(load all 84 compositions to match this statistic)
Mp00080: Set partitions āto permutationā¶ Permutations
St001665: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001665: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = -1 + 1
{{1,2}}
=> [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => 0 = -1 + 1
{{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => 0 = -1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = -1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => 1 = 0 + 1
Description
The number of pure excedances of a permutation.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
Matching statistic: St001729
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00080: Set partitions āto permutationā¶ Permutations
St001729: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001729: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = -1 + 1
{{1,2}}
=> [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => 0 = -1 + 1
{{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => 0 = -1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = -1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = -1 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => 1 = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 1 = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1 = 0 + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => 1 = 0 + 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
The following 367 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000290The major index of a binary word. St000291The number of descents of a binary word. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000662The staircase size of the code of a permutation. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000884The number of isolated descents of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001928The number of non-overlapping descents in a permutation. St000010The length of the partition. St000058The order of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000325The width of the tree associated to a permutation. St000346The number of coarsenings of a partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001642The Prague dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000142The number of even parts of a partition. St000147The largest part of an integer partition. St000156The Denert index of a permutation. St000157The number of descents of a standard tableau. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000340The number of non-final maximal constant sub-paths of length greater than one. St000347The inversion sum of a binary word. St000378The diagonal inversion number of an integer partition. St000387The matching number of a graph. St000390The number of runs of ones in a binary word. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000632The jump number of the poset. St000651The maximal size of a rise in a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000691The number of changes of a binary word. St000761The number of ascents in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree 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. St001071The beta invariant of the graph. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001393The induced matching number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001726The number of visible inversions of a permutation. St001743The discrepancy of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001792The arboricity of a graph. St001874Lusztig's a-function for the symmetric group. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000068The number of minimal elements in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000388The number of orbits of vertices of a graph under automorphisms. St000397The Strahler number of a rooted tree. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000527The width of the poset. St000630The length of the shortest palindromic decomposition of a binary word. St000638The number of up-down runs of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000740The last entry of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000764The number of strong records in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St000847The number of standard Young tableaux whose descent set is the binary word. St000903The number of different parts of an integer composition. St000918The 2-limited packing number of a graph. St000935The number of ordered refinements of an integer partition. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. 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:
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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001330The hat guessing number of a graph. St001389The number of partitions of the same length below the given integer partition. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000439The position of the first down step of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000659The number of rises of length at least 2 of a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000485The length of the longest cycle of a permutation. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000354The number of recoils of a permutation. St000389The number of runs of ones of odd length in a binary word. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000653The last descent of a permutation. St000658The number of rises of length 2 of a Dyck path. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000794The mak of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000874The position of the last double rise in a Dyck path. St000919The number of maximal left branches of a binary tree. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001114The number of odd descents of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001592The maximal number of simple paths between any two different vertices of a graph. St000061The number of nodes on the left branch of a binary tree. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000933The number of multipartitions of sizes given by an integer partition. St000990The first ascent of a permutation. St001062The maximal size of a block of a set partition. St001346The number of parking functions that give the same permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St000699The toughness times the least common multiple of 1,. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. 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. 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. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. 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 $eA$. St001352The number of internal nodes in the modular decomposition of a graph. St001500The global dimension of magnitude 1 Nakayama algebras. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St001118The acyclic chromatic index of a graph. 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 $eA$. St000806The semiperimeter of the associated bargraph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001570The minimal number of edges to add to make a graph Hamiltonian. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001060The distinguishing index of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001734The lettericity of a graph. St001569The maximal modular displacement of a permutation. St001555The order of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001870The number of positive entries followed by a negative entry in a signed permutation. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000456The monochromatic index of a connected graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Sorry, this statistic was not found in the database
or
add this statistic to the database ā it's very simple and we need your support!