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Matching statistic: St001824
St001824: Decorated permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => 0
[-] => 0
[+,+] => 0
[-,+] => 0
[+,-] => 0
[-,-] => 0
[2,1] => 0
[+,+,+] => 0
[-,+,+] => 0
[+,-,+] => 0
[+,+,-] => 0
[-,-,+] => 0
[-,+,-] => 0
[+,-,-] => 0
[-,-,-] => 0
[+,3,2] => 0
[-,3,2] => 0
[2,1,+] => 0
[2,1,-] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 0
[3,-,1] => 0
[+,+,+,+] => 0
[-,+,+,+] => 0
[+,-,+,+] => 0
[+,+,-,+] => 0
[+,+,+,-] => 0
[-,-,+,+] => 0
[-,+,-,+] => 0
[-,+,+,-] => 0
[+,-,-,+] => 0
[+,-,+,-] => 0
[+,+,-,-] => 0
[-,-,-,+] => 0
[-,-,+,-] => 0
[-,+,-,-] => 0
[+,-,-,-] => 0
[-,-,-,-] => 0
[+,+,4,3] => 0
[-,+,4,3] => 0
[+,-,4,3] => 0
[-,-,4,3] => 0
[+,3,2,+] => 0
[-,3,2,+] => 0
[+,3,2,-] => 0
[-,3,2,-] => 0
[+,3,4,2] => 0
[-,3,4,2] => 0
[+,4,2,3] => 0
Description
The number of crossed alignments in a decorated permutation.
Let $w$ be a decorated permutation.
An alignment $(i \mapsto w(i), j \mapsto w(j))$ is crossed if there exists a third arc $(h \mapsto w(h))$ which forms a crossing with both arcs in the alignment.
Matching statistic: St000405
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[+,-,+] => [1,3,2] => [3,1,2] => [3,2,1] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [3,1,2] => [3,2,1] => 0
[-,3,2] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[2,1,+] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[2,1,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,+,1] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,-,1] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[+,-,+,+] => [1,3,4,2] => [2,4,1,3] => [4,3,1,2] => 0
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,3,2,1] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [4,2,1,3] => 0
[-,+,-,+] => [2,4,1,3] => [1,3,4,2] => [1,4,2,3] => 0
[-,+,+,-] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[+,-,-,+] => [1,4,2,3] => [3,4,1,2] => [3,1,4,2] => 0
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[-,-,+,-] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [4,1,2,3] => [4,3,2,1] => 0
[-,+,4,3] => [2,4,1,3] => [1,3,4,2] => [1,4,2,3] => 0
[+,-,4,3] => [1,4,2,3] => [3,4,1,2] => [3,1,4,2] => 0
[-,-,4,3] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[+,3,2,+] => [1,3,4,2] => [2,4,1,3] => [4,3,1,2] => 0
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => [4,2,1,3] => 0
[+,3,2,-] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[-,3,2,-] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[+,3,4,2] => [1,4,2,3] => [3,4,1,2] => [3,1,4,2] => 0
[-,3,4,2] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[+,4,2,3] => [1,3,4,2] => [2,4,1,3] => [4,3,1,2] => 0
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St000516
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => ? ∊ {0,0}
[-] => [1] => [1] => [1] => ? ∊ {0,0}
[+,+] => [1,2] => [1,2] => [2,1] => 0
[-,+] => [1,2] => [1,2] => [2,1] => 0
[+,-] => [1,2] => [1,2] => [2,1] => 0
[-,-] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[-,+,+] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[+,-,+] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[-,-,+] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[-,+,-] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[+,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 0
[-,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 0
[2,1,+] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,1,-] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,3,1] => [2,3,1] => [3,2,1] => [2,1,3] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,+,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0
[3,-,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 0
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 0
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 0
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 0
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 0
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 0
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 0
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 0
[+,3,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 0
[-,3,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 0
[+,4,2,3] => [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 0
[-,4,2,3] => [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 0
[+,4,+,2] => [1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 0
Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St000319
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001280
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001587
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St001657
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St001918
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St000175
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[-,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[+,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[-,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[2,1] => [2,1] => [1,1]
=> [1]
=> 0
[+,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> 0
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> 0
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[+,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[-,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[+,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[+,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[+,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[-,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[-,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[-,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[+,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[+,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[+,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[-,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[-,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[-,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[+,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[-,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
[+,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[-,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[+,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[-,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[+,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[-,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[+,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[-,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,+,+] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,-,+] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,+,-] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,-,-] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,+] => [2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,1,-] => [2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,4,1] => [2,3,4,1] => [3,1]
=> [1]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,2,+] => [3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,2,-] => [3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [4,1,2,3] => [3,1]
=> [1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,+,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000478Another weight of a partition according to Alladi. 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. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001868The number of alignments of type NE of a signed permutation. St001866The nesting alignments of a signed permutation. St001490The number of connected components of a skew partition. St000068The number of minimal elements in a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000907The number of maximal antichains of minimal length in a poset. St001857The number of edges in the reduced word graph of a signed permutation. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
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