Your data matches 81 different statistics following compositions of up to 3 maps.
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St001687: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 2
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[2,3,4,5,1] => 0
[2,3,5,4,1] => 0
[2,4,3,5,1] => 1
[2,4,5,3,1] => 0
[2,5,3,4,1] => 0
[2,5,4,3,1] => 0
[3,2,4,5,1] => 1
[3,2,5,4,1] => 1
[3,4,2,5,1] => 2
[3,4,5,2,1] => 0
[3,5,2,4,1] => 1
[3,5,4,2,1] => 0
[4,2,3,5,1] => 1
[4,2,5,3,1] => 1
[4,3,2,5,1] => 2
[4,3,5,2,1] => 1
[4,5,2,3,1] => 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Mp00064: Permutations reversePermutations
St000358: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[1,3,2] => [2,3,1] => 0
[2,1,3] => [3,1,2] => 1
[2,3,1] => [1,3,2] => 0
[3,1,2] => [2,1,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => 0
[1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [2,4,3,1] => 0
[1,4,2,3] => [3,2,4,1] => 0
[1,4,3,2] => [2,3,4,1] => 0
[2,1,3,4] => [4,3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => 2
[2,3,4,1] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [1,3,4,2] => 0
[3,1,2,4] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [4,1,2,3] => 2
[3,2,4,1] => [1,4,2,3] => 1
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => 0
[4,1,3,2] => [2,3,1,4] => 0
[4,2,1,3] => [3,1,2,4] => 1
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[2,3,4,5,1] => [1,5,4,3,2] => 0
[2,3,5,4,1] => [1,4,5,3,2] => 0
[2,4,3,5,1] => [1,5,3,4,2] => 1
[2,4,5,3,1] => [1,3,5,4,2] => 0
[2,5,3,4,1] => [1,4,3,5,2] => 0
[2,5,4,3,1] => [1,3,4,5,2] => 0
[3,2,4,5,1] => [1,5,4,2,3] => 1
[3,2,5,4,1] => [1,4,5,2,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => 2
[3,4,5,2,1] => [1,2,5,4,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => 1
[3,5,4,2,1] => [1,2,4,5,3] => 0
[4,2,3,5,1] => [1,5,3,2,4] => 1
[4,2,5,3,1] => [1,3,5,2,4] => 1
[4,3,2,5,1] => [1,5,2,3,4] => 2
[4,3,5,2,1] => [1,2,5,3,4] => 1
[4,5,2,3,1] => [1,3,2,5,4] => 0
Description
The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Mp00069: Permutations complementPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
St000223: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => [3,2,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 0
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 0
[1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 0
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => [2,4,3,1] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 2
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => 2
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 0
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[2,3,5,4,1] => [4,3,1,2,5] => [3,1,4,2,5] => 0
[2,4,3,5,1] => [4,2,3,1,5] => [3,4,2,1,5] => 1
[2,4,5,3,1] => [4,2,1,3,5] => [2,4,1,3,5] => 0
[2,5,3,4,1] => [4,1,3,2,5] => [3,4,1,2,5] => 0
[2,5,4,3,1] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[3,2,4,5,1] => [3,4,2,1,5] => [2,4,3,1,5] => 1
[3,2,5,4,1] => [3,4,1,2,5] => [4,1,3,2,5] => 1
[3,4,2,5,1] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[3,4,5,2,1] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[3,5,2,4,1] => [3,1,4,2,5] => [4,3,1,2,5] => 1
[3,5,4,2,1] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[4,2,3,5,1] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[4,2,5,3,1] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[4,3,2,5,1] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[4,3,5,2,1] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => 0
Description
The number of nestings in the permutation.
Mp00064: Permutations reversePermutations
Mp00069: Permutations complementPermutations
St000356: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [2,1,3] => 0
[2,1,3] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [1,3,2] => [3,1,2] => 0
[3,1,2] => [2,1,3] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 1
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 0
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 0
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 0
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 1
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 2
[2,3,4,1] => [1,4,3,2] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [4,2,1,3] => 0
[3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 2
[3,2,4,1] => [1,4,2,3] => [4,1,3,2] => 1
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => 0
[3,4,2,1] => [1,2,4,3] => [4,3,1,2] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,4,1] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 0
[4,2,1,3] => [3,1,2,4] => [2,4,3,1] => 1
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => 0
[4,3,1,2] => [2,1,3,4] => [3,4,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,4,5,1] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[2,3,5,4,1] => [1,4,5,3,2] => [5,2,1,3,4] => 0
[2,4,3,5,1] => [1,5,3,4,2] => [5,1,3,2,4] => 1
[2,4,5,3,1] => [1,3,5,4,2] => [5,3,1,2,4] => 0
[2,5,3,4,1] => [1,4,3,5,2] => [5,2,3,1,4] => 0
[2,5,4,3,1] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[3,2,4,5,1] => [1,5,4,2,3] => [5,1,2,4,3] => 1
[3,2,5,4,1] => [1,4,5,2,3] => [5,2,1,4,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => [5,1,4,2,3] => 2
[3,4,5,2,1] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => [5,2,4,1,3] => 1
[3,5,4,2,1] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[4,2,3,5,1] => [1,5,3,2,4] => [5,1,3,4,2] => 1
[4,2,5,3,1] => [1,3,5,2,4] => [5,3,1,4,2] => 1
[4,3,2,5,1] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[4,3,5,2,1] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[4,5,2,3,1] => [1,3,2,5,4] => [5,3,4,1,2] => 0
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Mp00069: Permutations complementPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
St000371: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => [3,2,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 0
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 0
[1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 0
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => [2,4,3,1] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 2
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => 2
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 0
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[2,3,5,4,1] => [4,3,1,2,5] => [3,1,4,2,5] => 0
[2,4,3,5,1] => [4,2,3,1,5] => [3,4,2,1,5] => 1
[2,4,5,3,1] => [4,2,1,3,5] => [2,4,1,3,5] => 0
[2,5,3,4,1] => [4,1,3,2,5] => [3,4,1,2,5] => 0
[2,5,4,3,1] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[3,2,4,5,1] => [3,4,2,1,5] => [2,4,3,1,5] => 1
[3,2,5,4,1] => [3,4,1,2,5] => [4,1,3,2,5] => 1
[3,4,2,5,1] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[3,4,5,2,1] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[3,5,2,4,1] => [3,1,4,2,5] => [4,3,1,2,5] => 1
[3,5,4,2,1] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[4,2,3,5,1] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[4,2,5,3,1] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[4,3,2,5,1] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[4,3,5,2,1] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => 0
Description
The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].
Mp00069: Permutations complementPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00239: Permutations CorteelPermutations
St000039: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [3,2,1] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [4,2,3,1] => 0
[1,2,4,3] => [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 0
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 1
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 0
[1,4,2,3] => [4,1,3,2] => [3,4,1,2] => [4,3,2,1] => 0
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => [2,4,3,1] => [3,2,4,1] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [3,1,4,2] => 1
[2,3,1,4] => [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 2
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [3,4,2,1] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => [3,2,4,1] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [4,3,2,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 0
[2,3,5,4,1] => [4,3,1,2,5] => [3,1,4,2,5] => [4,1,3,2,5] => 0
[2,4,3,5,1] => [4,2,3,1,5] => [3,4,2,1,5] => [4,3,1,2,5] => 1
[2,4,5,3,1] => [4,2,1,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => 0
[2,5,3,4,1] => [4,1,3,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 0
[2,5,4,3,1] => [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[3,2,4,5,1] => [3,4,2,1,5] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[3,2,5,4,1] => [3,4,1,2,5] => [4,1,3,2,5] => [3,1,4,2,5] => 1
[3,4,2,5,1] => [3,2,4,1,5] => [4,3,2,1,5] => [3,4,1,2,5] => 2
[3,4,5,2,1] => [3,2,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 0
[3,5,2,4,1] => [3,1,4,2,5] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[3,5,4,2,1] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[4,2,3,5,1] => [2,4,3,1,5] => [3,2,4,1,5] => [2,4,3,1,5] => 1
[4,2,5,3,1] => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 1
[4,3,2,5,1] => [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 2
[4,3,5,2,1] => [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
Description
The number of crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Mp00069: Permutations complementPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00064: Permutations reversePermutations
St000372: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [1,3,2] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [2,1,3] => 0
[2,1,3] => [2,3,1] => [3,2,1] => [1,2,3] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [3,1,2] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [1,4,3,2] => 0
[1,2,4,3] => [4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 0
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [1,2,4,3] => 1
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 0
[1,4,2,3] => [4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 0
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [3,2,1,4] => 0
[2,1,3,4] => [3,4,2,1] => [2,4,3,1] => [1,3,4,2] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 1
[2,3,1,4] => [3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 2
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [2,1,3,4] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 0
[3,1,2,4] => [2,4,3,1] => [3,2,4,1] => [1,4,2,3] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,1,2,4] => 1
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 2
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 0
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,4,5,1] => [4,3,2,1,5] => [2,3,4,1,5] => [5,1,4,3,2] => 0
[2,3,5,4,1] => [4,3,1,2,5] => [3,1,4,2,5] => [5,2,4,1,3] => 0
[2,4,3,5,1] => [4,2,3,1,5] => [3,4,2,1,5] => [5,1,2,4,3] => 1
[2,4,5,3,1] => [4,2,1,3,5] => [2,4,1,3,5] => [5,3,1,4,2] => 0
[2,5,3,4,1] => [4,1,3,2,5] => [3,4,1,2,5] => [5,2,1,4,3] => 0
[2,5,4,3,1] => [4,1,2,3,5] => [4,1,2,3,5] => [5,3,2,1,4] => 0
[3,2,4,5,1] => [3,4,2,1,5] => [2,4,3,1,5] => [5,1,3,4,2] => 1
[3,2,5,4,1] => [3,4,1,2,5] => [4,1,3,2,5] => [5,2,3,1,4] => 1
[3,4,2,5,1] => [3,2,4,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 2
[3,4,5,2,1] => [3,2,1,4,5] => [2,3,1,4,5] => [5,4,1,3,2] => 0
[3,5,2,4,1] => [3,1,4,2,5] => [4,3,1,2,5] => [5,2,1,3,4] => 1
[3,5,4,2,1] => [3,1,2,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => 0
[4,2,3,5,1] => [2,4,3,1,5] => [3,2,4,1,5] => [5,1,4,2,3] => 1
[4,2,5,3,1] => [2,4,1,3,5] => [4,2,1,3,5] => [5,3,1,2,4] => 1
[4,3,2,5,1] => [2,3,4,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => 2
[4,3,5,2,1] => [2,3,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 1
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 0
Description
The number of mid points of increasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$. The generating function is given by [1].
Mp00064: Permutations reversePermutations
Mp00326: Permutations weak order rowmotionPermutations
Mp00066: Permutations inversePermutations
St001683: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,3,2] => [2,3,1] => [3,1,2] => 0
[3,1,2] => [2,1,3] => [3,1,2] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [3,1,2,4] => [2,3,1,4] => 0
[1,3,2,4] => [4,2,3,1] => [2,4,1,3] => [3,1,4,2] => 1
[1,3,4,2] => [2,4,3,1] => [2,1,3,4] => [2,1,3,4] => 0
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => [3,1,2,4] => 0
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 0
[2,1,3,4] => [4,3,1,2] => [1,3,4,2] => [1,4,2,3] => 1
[2,1,4,3] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 1
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => [1,3,4,2] => 2
[2,3,4,1] => [1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [1,3,4,2] => [3,2,4,1] => [4,2,1,3] => 0
[3,1,2,4] => [4,2,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,4,2] => [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 1
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,4,1] => [1,4,2,3] => [2,4,3,1] => [4,1,3,2] => 1
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 0
[4,1,2,3] => [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 0
[4,1,3,2] => [2,3,1,4] => [4,2,1,3] => [3,2,4,1] => 0
[4,2,1,3] => [3,1,2,4] => [4,1,3,2] => [2,4,3,1] => 1
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[2,3,4,5,1] => [1,5,4,3,2] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[2,3,5,4,1] => [1,4,5,3,2] => [4,2,3,5,1] => [5,2,3,1,4] => 0
[2,4,3,5,1] => [1,5,3,4,2] => [3,5,2,4,1] => [5,3,1,4,2] => 1
[2,4,5,3,1] => [1,3,5,4,2] => [3,2,4,5,1] => [5,2,1,3,4] => 0
[2,5,3,4,1] => [1,4,3,5,2] => [3,4,2,5,1] => [5,3,1,2,4] => 0
[2,5,4,3,1] => [1,3,4,5,2] => [4,3,2,5,1] => [5,3,2,1,4] => 0
[3,2,4,5,1] => [1,5,4,2,3] => [2,4,5,3,1] => [5,1,4,2,3] => 1
[3,2,5,4,1] => [1,4,5,2,3] => [4,2,5,3,1] => [5,2,4,1,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => [2,5,3,4,1] => [5,1,3,4,2] => 2
[3,4,5,2,1] => [1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => [2,4,3,5,1] => [5,1,3,2,4] => 1
[3,5,4,2,1] => [1,2,4,5,3] => [4,3,5,2,1] => [5,4,2,1,3] => 0
[4,2,3,5,1] => [1,5,3,2,4] => [2,3,5,4,1] => [5,1,2,4,3] => 1
[4,2,5,3,1] => [1,3,5,2,4] => [3,2,5,4,1] => [5,2,1,4,3] => 1
[4,3,2,5,1] => [1,5,2,3,4] => [2,5,4,3,1] => [5,1,4,3,2] => 2
[4,3,5,2,1] => [1,2,5,3,4] => [3,5,4,2,1] => [5,4,1,3,2] => 1
[4,5,2,3,1] => [1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => 0
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001866: Signed permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [3,2,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [3,2,4,1] => 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 2
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => [1,3,4,2] => 0
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => [4,3,1,2] => 1
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => [3,4,2,1] => 1
[3,1,4,2] => [4,3,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 1
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => [3,1,2,4] => 0
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[2,3,5,4,1] => [4,2,3,5,1] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[2,4,3,5,1] => [5,2,4,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[2,4,5,3,1] => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[2,5,3,4,1] => [4,2,5,3,1] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[2,5,4,3,1] => [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[3,2,4,5,1] => [5,3,2,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[3,2,5,4,1] => [4,3,2,5,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[3,4,2,5,1] => [5,4,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,4,5,2,1] => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[3,5,2,4,1] => [4,5,3,2,1] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[3,5,4,2,1] => [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[4,2,3,5,1] => [5,4,2,3,1] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[4,2,5,3,1] => [3,5,4,2,1] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[4,3,2,5,1] => [5,3,4,2,1] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[4,3,5,2,1] => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[4,5,2,3,1] => [3,5,2,4,1] => [1,4,2,5,3] => [1,4,2,5,3] => 0
Description
The nesting alignments of a signed permutation. A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that * $-i < -j < -\pi(j) < -\pi(i)$, or * $-i < j \leq \pi(j) < -\pi(i)$, or * $i < j \leq \pi(j) < \pi(i)$.
Mp00064: Permutations reversePermutations
Mp00066: Permutations inversePermutations
Mp00170: Permutations to signed permutationSigned permutations
St001882: Signed permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [3,1,2] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 0
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 0
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [2,4,3,1] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 2
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[2,4,5,3,1] => [1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[2,5,3,4,1] => [1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[2,5,4,3,1] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[3,2,4,5,1] => [1,5,4,2,3] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[3,2,5,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[3,5,4,2,1] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[4,2,3,5,1] => [1,5,3,2,4] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The number of occurrences of a type-B 231 pattern in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
The following 71 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St001857The number of edges in the reduced word graph of a signed permutation. St000091The descent variation of a composition. St000562The number of internal points of a set partition. St001550The number of inversions between exceedances where the greater exceedance is linked. St001715The number of non-records in a permutation. St001867The number of alignments of type EN of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000284The Plancherel distribution on integer partitions. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001568The smallest positive integer that does not appear twice in the partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001875The number of simple modules with projective dimension at most 1. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.