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Matching statistic: St000034
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Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000034: 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] => 0
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [1,2] => 0
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 0
[+,-,+] => [1,3,2] => 0
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 0
[-,+,-] => [2,1,3] => 0
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,2,3] => 0
[-,3,2] => [2,1,3] => 0
[2,1,+] => [1,3,2] => 0
[2,1,-] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,+,1] => [2,1,3] => 0
[3,-,1] => [1,3,2] => 0
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 0
[+,-,+,+] => [1,3,4,2] => 0
[+,+,-,+] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 1
[-,+,-,+] => [2,4,1,3] => 0
[-,+,+,-] => [2,3,1,4] => 0
[+,-,-,+] => [1,4,2,3] => 0
[+,-,+,-] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 0
[-,-,+,-] => [3,1,2,4] => 0
[-,+,-,-] => [2,1,3,4] => 0
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => 0
[+,-,4,3] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => 0
[+,3,2,+] => [1,2,4,3] => 0
[-,3,2,+] => [2,4,1,3] => 0
[+,3,2,-] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => 0
[+,3,4,2] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => 0
[+,4,2,3] => [1,2,3,4] => 0
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St000360
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000360: 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
Mp00066: Permutations —inverse⟶ Permutations
St000360: 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] => [1,2] => [1,2] => [1,2] => 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] => [2,3,1] => 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] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,1,+] => [1,3,2] => [3,1,2] => [2,3,1] => 0
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,-,1] => [1,3,2] => [3,1,2] => [2,3,1] => 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] => [3,1,4,2] => 0
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 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] => 0
[+,-,-,+] => [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 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] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,-,4,3] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[-,-,4,3] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 0
[-,3,2,+] => [2,4,1,3] => [1,3,4,2] => [1,4,2,3] => 0
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of occurrences of the pattern 32-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $32\!\!-\!\!1$.
Matching statistic: St000375
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000375: 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
[-,+] => [1,2] => [1,2] => [1,2] => 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] => [3,2,1] => [3,2,1] => 0
[-,+,+] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[+,-,+] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[+,+,-] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[-,-,+] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[-,+,-] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[+,-,-] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[-,-,-] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[+,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 0
[-,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 0
[2,1,+] => [2,1,3] => [3,1,2] => [1,3,2] => 0
[2,1,-] => [2,1,3] => [3,1,2] => [1,3,2] => 0
[2,3,1] => [2,3,1] => [1,3,2] => [2,3,1] => 0
[3,1,2] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[3,+,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[3,-,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[+,+,+,+] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[-,+,+,+] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[+,-,+,+] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[+,+,-,+] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[+,+,+,-] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[-,-,+,+] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[-,+,-,+] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[-,+,+,-] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[+,-,-,+] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[+,-,+,-] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[+,+,-,-] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[-,-,-,+] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[-,-,+,-] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[-,+,-,-] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[+,-,-,-] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[-,-,-,-] => [1,2,3,4] => [4,2,3,1] => [4,1,3,2] => 0
[+,+,4,3] => [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 0
[-,+,4,3] => [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 0
[+,-,4,3] => [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 0
[-,-,4,3] => [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 0
[+,3,2,+] => [1,3,2,4] => [2,4,1,3] => [1,3,4,2] => 0
[-,3,2,+] => [1,3,2,4] => [2,4,1,3] => [1,3,4,2] => 0
[+,3,2,-] => [1,3,2,4] => [2,4,1,3] => [1,3,4,2] => 0
[-,3,2,-] => [1,3,2,4] => [2,4,1,3] => [1,3,4,2] => 0
[+,3,4,2] => [1,3,4,2] => [2,3,4,1] => [4,1,2,3] => 0
[-,3,4,2] => [1,3,4,2] => [2,3,4,1] => [4,1,2,3] => 0
[+,4,2,3] => [1,4,2,3] => [3,4,1,2] => [1,4,2,3] => 0
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St001513
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001513: 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
[-,+] => [1,2] => [1,2] => [1,2] => 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] => [3,2,1] => [3,2,1] => 0
[-,+,+] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[+,-,+] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[+,+,-] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[-,-,+] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[-,+,-] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[+,-,-] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[-,-,-] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[+,3,2] => [1,3,2] => [2,3,1] => [2,3,1] => 0
[-,3,2] => [1,3,2] => [2,3,1] => [2,3,1] => 0
[2,1,+] => [2,1,3] => [3,1,2] => [1,3,2] => 0
[2,1,-] => [2,1,3] => [3,1,2] => [1,3,2] => 0
[2,3,1] => [2,3,1] => [1,3,2] => [3,1,2] => 0
[3,1,2] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[3,+,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[3,-,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[+,+,+,+] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[-,+,+,+] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[+,-,+,+] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[+,+,-,+] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[+,+,+,-] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[-,-,+,+] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[-,+,-,+] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[-,+,+,-] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[+,-,-,+] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[+,-,+,-] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[+,+,-,-] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[-,-,-,+] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[-,-,+,-] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[-,+,-,-] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[+,-,-,-] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[-,-,-,-] => [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 0
[+,+,4,3] => [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 0
[-,+,4,3] => [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 0
[+,-,4,3] => [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 0
[-,-,4,3] => [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 0
[+,3,2,+] => [1,3,2,4] => [2,4,1,3] => [2,1,4,3] => 0
[-,3,2,+] => [1,3,2,4] => [2,4,1,3] => [2,1,4,3] => 0
[+,3,2,-] => [1,3,2,4] => [2,4,1,3] => [2,1,4,3] => 0
[-,3,2,-] => [1,3,2,4] => [2,4,1,3] => [2,1,4,3] => 0
[+,3,4,2] => [1,3,4,2] => [2,3,4,1] => [2,3,4,1] => 0
[-,3,4,2] => [1,3,4,2] => [2,3,4,1] => [2,3,4,1] => 0
[+,4,2,3] => [1,4,2,3] => [3,4,1,2] => [1,3,4,2] => 0
Description
The number of nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St000175
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: 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
St000175: 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]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
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:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000225
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: 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
St000225: 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]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
Difference between largest and smallest parts in a partition.
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]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
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]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2}
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: St000749
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: 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
St000749: 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]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The 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.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St000944
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: 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
St000944: 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]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
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The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001280The number of parts of an integer partition that are at least two. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000478Another weight of a partition according to Alladi. 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. St001862The number of crossings of a signed permutation. 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. St001866The nesting alignments of a signed permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001845The number of join irreducibles minus the rank of a lattice. 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. 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. St001768The number of reduced words of a signed permutation. 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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