Your data matches 44 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000662
Mp00256: Decorated permutations upper permutationPermutations
St000662: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 1
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 1
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => 1
[+,-,4,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => 1
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Mp00256: Decorated permutations upper permutationPermutations
St000994: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 1
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 1
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => 1
[+,-,4,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => 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. See [2] for the exponential generating function, also see [3].
Mp00256: Decorated permutations upper permutationPermutations
St001761: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 1
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 1
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => 1
[+,-,4,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => 1
Description
The maximal multiplicity of a letter in a reduced word of a permutation. For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Mp00256: Decorated permutations upper permutationPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
St000021: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [1,3,2] => 1
[+,-,+] => [1,3,2] => [3,1,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [2,3,1] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [3,1,2] => 1
[-,3,2] => [3,1,2] => [2,3,1] => 1
[2,1,+] => [2,3,1] => [1,3,2] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [2,3,1] => 1
[3,1,2] => [2,3,1] => [1,3,2] => 1
[3,+,1] => [2,3,1] => [1,3,2] => 1
[3,-,1] => [3,1,2] => [2,3,1] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [1,2,4,3] => 1
[+,-,+,+] => [1,3,4,2] => [2,4,1,3] => 1
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => 2
[-,+,-,+] => [2,4,1,3] => [1,3,4,2] => 1
[-,+,+,-] => [2,3,1,4] => [1,3,2,4] => 1
[+,-,-,+] => [1,4,2,3] => [3,4,1,2] => 1
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [2,3,4,1] => 1
[-,-,+,-] => [3,1,2,4] => [2,3,1,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[-,+,4,3] => [2,4,1,3] => [1,3,4,2] => 1
[+,-,4,3] => [1,4,2,3] => [3,4,1,2] => 1
[-,-,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[+,3,2,+] => [1,3,4,2] => [2,4,1,3] => 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => 2
[+,3,2,-] => [1,3,2,4] => [3,1,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [2,3,1,4] => 1
[+,3,4,2] => [1,4,2,3] => [3,4,1,2] => 1
[-,3,4,2] => [4,1,2,3] => [2,3,4,1] => 1
[+,4,2,3] => [1,3,4,2] => [2,4,1,3] => 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Mp00256: Decorated permutations upper permutationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
St000035: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,1,2] => 1
[+,-,+] => [1,3,2] => [1,3,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,2,1] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => [3,2,1] => 1
[2,1,+] => [2,3,1] => [3,1,2] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,2,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => 1
[3,+,1] => [2,3,1] => [3,1,2] => 1
[3,-,1] => [3,1,2] => [3,2,1] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => 1
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => 2
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => 1
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => 1
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => [4,3,1,2] => 1
[+,-,4,3] => [1,4,2,3] => [1,4,3,2] => 1
[-,-,4,3] => [4,1,2,3] => [4,3,2,1] => 1
[+,3,2,+] => [1,3,4,2] => [1,4,2,3] => 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [3,2,1,4] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,3,2] => 1
[-,3,4,2] => [4,1,2,3] => [4,3,2,1] => 1
[+,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
Description
The number of left outer peaks of a permutation. A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$. In other words, it is a peak in the word $[0,w_1,..., w_n]$. This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Mp00256: Decorated permutations upper permutationPermutations
Mp00239: Permutations CorteelPermutations
St000155: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,2,1] => 1
[+,-,+] => [1,3,2] => [1,3,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,1,2] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => [3,2,1] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [3,2,1] => 1
[3,+,1] => [2,3,1] => [3,2,1] => 1
[3,-,1] => [3,1,2] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => 1
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [4,3,2,1] => 2
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => 1
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => [4,2,1,3] => 1
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => [1,4,3,2] => 1
[-,3,2,+] => [3,4,1,2] => [4,3,2,1] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Mp00256: Decorated permutations upper permutationPermutations
Mp00239: Permutations CorteelPermutations
St000162: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,2,1] => 1
[+,-,+] => [1,3,2] => [1,3,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,1,2] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => [3,2,1] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [3,2,1] => 1
[3,+,1] => [2,3,1] => [3,2,1] => 1
[3,-,1] => [3,1,2] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => 1
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [4,3,2,1] => 2
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => 1
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => [4,2,1,3] => 1
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => [1,4,3,2] => 1
[-,3,2,+] => [3,4,1,2] => [4,3,2,1] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => [1,4,3,2] => 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]]).
Mp00256: Decorated permutations upper permutationPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
St000884: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [1,3,2] => 1
[+,-,+] => [1,3,2] => [3,1,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [2,3,1] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [3,1,2] => 1
[-,3,2] => [3,1,2] => [2,3,1] => 1
[2,1,+] => [2,3,1] => [1,3,2] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [2,3,1] => 1
[3,1,2] => [2,3,1] => [1,3,2] => 1
[3,+,1] => [2,3,1] => [1,3,2] => 1
[3,-,1] => [3,1,2] => [2,3,1] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [1,2,4,3] => 1
[+,-,+,+] => [1,3,4,2] => [2,4,1,3] => 1
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => 2
[-,+,-,+] => [2,4,1,3] => [1,3,4,2] => 1
[-,+,+,-] => [2,3,1,4] => [1,3,2,4] => 1
[+,-,-,+] => [1,4,2,3] => [3,4,1,2] => 1
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [2,3,4,1] => 1
[-,-,+,-] => [3,1,2,4] => [2,3,1,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[-,+,4,3] => [2,4,1,3] => [1,3,4,2] => 1
[+,-,4,3] => [1,4,2,3] => [3,4,1,2] => 1
[-,-,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[+,3,2,+] => [1,3,4,2] => [2,4,1,3] => 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => 2
[+,3,2,-] => [1,3,2,4] => [3,1,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [2,3,1,4] => 1
[+,3,4,2] => [1,4,2,3] => [3,4,1,2] => 1
[-,3,4,2] => [4,1,2,3] => [2,3,4,1] => 1
[+,4,2,3] => [1,3,4,2] => [2,4,1,3] => 1
Description
The number of isolated descents of a permutation. A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Mp00256: Decorated permutations upper permutationPermutations
Mp00239: Permutations CorteelPermutations
St001269: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,2,1] => 1
[+,-,+] => [1,3,2] => [1,3,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,1,2] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => [3,2,1] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [3,2,1] => 1
[3,+,1] => [2,3,1] => [3,2,1] => 1
[3,-,1] => [3,1,2] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => 1
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [4,3,2,1] => 2
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => 1
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => [4,2,1,3] => 1
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => [1,4,3,2] => 1
[-,3,2,+] => [3,4,1,2] => [4,3,2,1] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St001277
Mp00256: Decorated permutations upper permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
St001277: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 0
[-] => [1] => ([],1)
=> 0
[+,+] => [1,2] => ([],2)
=> 0
[-,+] => [2,1] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => ([],2)
=> 0
[-,-] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> 1
[+,+,+] => [1,2,3] => ([],3)
=> 0
[-,+,+] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[+,-,+] => [1,3,2] => ([(1,2)],3)
=> 1
[+,+,-] => [1,2,3] => ([],3)
=> 0
[-,-,+] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[-,+,-] => [2,1,3] => ([(1,2)],3)
=> 1
[+,-,-] => [1,2,3] => ([],3)
=> 0
[-,-,-] => [1,2,3] => ([],3)
=> 0
[+,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[-,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[2,1,+] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,-] => [2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,+,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,-,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[+,+,+,+] => [1,2,3,4] => ([],4)
=> 0
[-,+,+,+] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,-,+,+] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[+,+,-,+] => [1,2,4,3] => ([(2,3)],4)
=> 1
[+,+,+,-] => [1,2,3,4] => ([],4)
=> 0
[-,-,+,+] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[-,+,+,-] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[+,-,-,+] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[+,-,+,-] => [1,3,2,4] => ([(2,3)],4)
=> 1
[+,+,-,-] => [1,2,3,4] => ([],4)
=> 0
[-,-,-,+] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[-,-,+,-] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[-,+,-,-] => [2,1,3,4] => ([(2,3)],4)
=> 1
[+,-,-,-] => [1,2,3,4] => ([],4)
=> 0
[-,-,-,-] => [1,2,3,4] => ([],4)
=> 0
[+,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[-,+,4,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[+,-,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[-,-,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,3,2,+] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[-,3,2,+] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[+,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> 1
[-,3,2,-] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[+,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[-,3,4,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
Description
The degeneracy of a graph. The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000451The length of the longest pattern of the form k 1 2. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000216The absolute length of a permutation. St000354The number of recoils of a permutation. St000390The number of runs of ones in a binary word. St000809The reduced reflection length of the permutation. St000702The number of weak deficiencies of a permutation. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001864The number of excedances of a signed permutation. St000456The monochromatic index of a connected graph. St001823The Stasinski-Voll length of a signed permutation.