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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000001
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Mp00256: Decorated permutations —upper permutation⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 1
[-] => [1] => 1
[+,+] => [1,2] => 1
[-,+] => [2,1] => 1
[+,-] => [1,2] => 1
[-,-] => [1,2] => 1
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 1
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 1
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 1
[-,-,-] => [1,2,3] => 1
[+,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => 1
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 1
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 1
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => 2
[+,-,4,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => 1
Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Matching statistic: St000880
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
St000880: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
St000880: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ? = 1
[-] => [1] => ? = 1
[+,+] => [1,2] => 1
[-,+] => [2,1] => 1
[+,-] => [1,2] => 1
[-,-] => [1,2] => 1
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 1
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 1
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 1
[-,-,-] => [1,2,3] => 1
[+,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => 1
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 1
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 1
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => 2
[+,-,4,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => 1
[-,4,2,3] => [3,4,1,2] => 2
[+,4,+,2] => [1,3,4,2] => 1
[-,-,-,+,+,+] => [4,5,6,1,2,3] => ? = 42
[-,-,-,-,+,+] => [5,6,1,2,3,4] => ? = 14
[-,-,-,+,-,+] => [4,6,1,2,3,5] => ? = 14
[-,-,-,+,+,-] => [4,5,1,2,3,6] => ? = 5
[-,-,-,-,-,+] => [6,1,2,3,4,5] => ? = 1
[-,-,-,-,+,-] => [5,1,2,3,4,6] => ? = 1
[-,-,-,+,-,-] => [4,1,2,3,5,6] => ? = 1
[-,-,-,+,6,5] => [4,6,1,2,3,5] => ? = 14
[-,-,-,-,6,5] => [6,1,2,3,4,5] => ? = 1
[-,-,-,5,4,+] => [5,6,1,2,3,4] => ? = 14
[-,-,-,5,4,-] => [5,1,2,3,4,6] => ? = 1
[-,-,-,5,6,4] => [6,1,2,3,4,5] => ? = 1
[-,-,-,6,4,5] => [5,6,1,2,3,4] => ? = 14
[-,-,-,6,+,4] => [5,6,1,2,3,4] => ? = 14
[-,-,-,6,-,4] => [6,1,2,3,4,5] => ? = 1
[-,-,4,3,+,+] => [4,5,6,1,2,3] => ? = 42
[-,-,4,3,-,+] => [4,6,1,2,3,5] => ? = 14
[-,-,4,3,+,-] => [4,5,1,2,3,6] => ? = 5
[-,-,4,3,-,-] => [4,1,2,3,5,6] => ? = 1
[-,-,4,3,6,5] => [4,6,1,2,3,5] => ? = 14
[-,-,4,5,3,+] => [5,6,1,2,3,4] => ? = 14
[-,-,4,5,3,-] => [5,1,2,3,4,6] => ? = 1
[-,-,4,5,6,3] => [6,1,2,3,4,5] => ? = 1
[-,-,4,6,3,5] => [5,6,1,2,3,4] => ? = 14
[-,-,4,6,+,3] => [5,6,1,2,3,4] => ? = 14
[-,-,4,6,-,3] => [6,1,2,3,4,5] => ? = 1
[-,-,5,3,4,+] => [4,5,6,1,2,3] => ? = 42
[-,-,5,3,4,-] => [4,5,1,2,3,6] => ? = 5
[-,-,5,3,6,4] => [4,6,1,2,3,5] => ? = 14
[-,-,5,+,3,+] => [4,5,6,1,2,3] => ? = 42
[-,-,5,-,3,+] => [5,6,1,2,3,4] => ? = 14
[-,-,5,+,3,-] => [4,5,1,2,3,6] => ? = 5
[-,-,5,-,3,-] => [5,1,2,3,4,6] => ? = 1
[-,-,5,+,6,3] => [4,6,1,2,3,5] => ? = 14
[-,-,5,-,6,3] => [6,1,2,3,4,5] => ? = 1
[-,-,5,6,3,4] => [5,6,1,2,3,4] => ? = 14
[-,-,5,6,4,3] => [5,6,1,2,3,4] => ? = 14
[-,-,6,3,4,5] => [4,5,6,1,2,3] => ? = 42
[-,-,6,3,+,4] => [4,5,6,1,2,3] => ? = 42
[-,-,6,3,-,4] => [4,6,1,2,3,5] => ? = 14
[-,-,6,+,3,5] => [4,5,6,1,2,3] => ? = 42
[-,-,6,-,3,5] => [5,6,1,2,3,4] => ? = 14
[-,-,6,+,+,3] => [4,5,6,1,2,3] => ? = 42
[-,-,6,-,+,3] => [5,6,1,2,3,4] => ? = 14
[-,-,6,+,-,3] => [4,6,1,2,3,5] => ? = 14
[-,-,6,-,-,3] => [6,1,2,3,4,5] => ? = 1
[-,-,6,5,3,4] => [5,6,1,2,3,4] => ? = 14
[-,-,6,5,4,3] => [5,6,1,2,3,4] => ? = 14
Description
The number of connected components of long braid edges in the graph of braid moves of a permutation.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a long braid move if they are obtained from each other by a modification of the form $s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2s_3$ and $s_1s_2s_3s_2$ for
$$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$$
share an edge because they are obtained from each other by interchanging $s_3s_2s_3 \leftrightarrow s_3s_2s_3$.
This statistic counts the number connected components of such long braid moves among all reduced words.
Matching statistic: St001862
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 18%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 18%
Values
[+] => [1] => [1] => [1] => 0 = 1 - 1
[-] => [1] => [1] => [1] => 0 = 1 - 1
[+,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,+] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[+,-,+] => [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[-,3,2] => [3,1,2] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[2,1,+] => [2,3,1] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[2,1,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[2,3,1] => [3,1,2] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[3,1,2] => [2,3,1] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[3,+,1] => [2,3,1] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[3,-,1] => [3,1,2] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => [4,3,1,2] => 1 = 2 - 1
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[-,+,4,3] => [2,4,1,3] => [4,3,1,2] => [4,3,1,2] => 1 = 2 - 1
[+,-,4,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[-,-,4,3] => [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[+,3,2,+] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[-,3,2,-] => [3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[+,3,4,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[-,3,4,2] => [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[+,4,2,3] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[-,+,+,+,+] => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,-,+,+,+] => [3,4,5,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 5 - 1
[-,+,-,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 5 - 1
[-,+,+,-,+] => [2,3,5,1,4] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 3 - 1
[-,+,+,+,-] => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[-,-,-,+,+] => [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 5 - 1
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 5 - 1
[-,-,+,+,-] => [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[-,+,-,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3 - 1
[-,+,-,+,-] => [2,4,1,3,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2 - 1
[-,+,+,-,-] => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 - 1
[-,-,-,-,+] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 - 1
[-,-,-,+,-] => [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[-,-,+,-,-] => [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 - 1
[-,+,-,-,-] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[-,+,+,5,4] => [2,3,5,1,4] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 3 - 1
[-,-,+,5,4] => [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 5 - 1
[-,+,-,5,4] => [2,5,1,3,4] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3 - 1
[-,-,-,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 - 1
[-,+,4,3,+] => [2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 5 - 1
[-,-,4,3,+] => [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 5 - 1
[-,+,4,3,-] => [2,4,1,3,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2 - 1
[-,-,4,3,-] => [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[-,+,4,5,3] => [2,5,1,3,4] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3 - 1
[-,-,4,5,3] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 - 1
[-,+,5,3,4] => [2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 5 - 1
[-,-,5,3,4] => [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 5 - 1
[-,+,5,+,3] => [2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 5 - 1
[-,-,5,+,3] => [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 5 - 1
[-,+,5,-,3] => [2,5,1,3,4] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3 - 1
[-,-,5,-,3] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 - 1
[-,3,2,+,+] => [3,4,5,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 5 - 1
[-,3,2,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 5 - 1
[-,3,2,+,-] => [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[-,3,2,-,-] => [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 - 1
[-,3,2,5,4] => [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 5 - 1
[-,3,4,2,+] => [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 5 - 1
[-,3,4,2,-] => [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[-,3,4,5,2] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 - 1
[-,3,5,2,4] => [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 5 - 1
[-,3,5,+,2] => [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 5 - 1
[-,3,5,-,2] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 - 1
[-,4,2,3,+] => [3,4,5,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 5 - 1
[-,4,2,3,-] => [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[-,4,2,5,3] => [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 5 - 1
[-,4,+,2,+] => [3,4,5,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 5 - 1
[-,4,-,2,+] => [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 5 - 1
[-,4,+,2,-] => [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[-,4,-,2,-] => [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[-,4,+,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 5 - 1
Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
* $i < j \leq \pi(i) < \pi(j)$, or
* $-i < j \leq -\pi(i) < \pi(j)$, or
* $i > j > \pi(i) > \pi(j)$.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[+] => [1] => [1] => ([],1)
=> ? = 1
[-] => [1] => [1] => ([],1)
=> ? = 1
[+,+] => [1,2] => [2] => ([],2)
=> ? = 1
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => [2] => ([],2)
=> ? = 1
[-,-] => [1,2] => [2] => ([],2)
=> ? = 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[+,+,+] => [1,2,3] => [3] => ([],3)
=> ? = 1
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[+,+,-] => [1,2,3] => [3] => ([],3)
=> ? = 1
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 1
[+,-,-] => [1,2,3] => [3] => ([],3)
=> ? = 1
[-,-,-] => [1,2,3] => [3] => ([],3)
=> ? = 1
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,-,1] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
=> ? = 1
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,+,+,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 1
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 1
[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 1
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 1
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 1
[+,-,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 1
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 1
[+,+,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[-,+,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
[-,-,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 1
[+,3,2,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[-,3,2,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 1
[+,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
[-,3,4,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 1
[+,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[-,4,2,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[+,4,+,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[-,4,+,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
[-,4,-,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 1
[2,1,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 1
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 1
[2,4,1,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,4,+,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,4,-,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 1
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,+,1,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,+,2] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,+,1,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,+,+,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,-,+,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,+,-,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,+,+,-,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,+,+,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,+,4,3,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,+,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,+,5,+,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,3,2,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,4,2,3,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,4,+,2,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,5,2,+,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,5,+,2,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[+,5,+,+,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[3,1,2,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[3,+,1,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,1,2,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,1,+,2,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,+,1,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,+,+,1,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[5,1,2,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[5,1,+,2,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[5,1,+,+,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[5,+,1,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[5,+,1,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001768
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001768: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 18%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001768: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 18%
Values
[+] => [1] => [1] => 1
[-] => [1] => [1] => 1
[+,+] => [1,2] => [1,2] => 1
[-,+] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => 1
[-,-] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => [1,2,3] => 1
[-,+,+] => [2,3,1] => [2,3,1] => 1
[+,-,+] => [1,3,2] => [1,3,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 1
[-,-,+] => [3,1,2] => [3,1,2] => 1
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 1
[-,-,-] => [1,2,3] => [1,2,3] => 1
[+,3,2] => [1,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => [2,3,1] => 1
[2,1,-] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [2,3,1] => 1
[3,+,1] => [2,3,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 1
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 1
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 1
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => [2,4,1,3] => 2
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => 1
[-,3,2,-] => [3,1,2,4] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => [1,3,4,2] => 1
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1
[-,+,+,+,+] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[+,-,+,+,+] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[+,+,-,+,+] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
[+,+,+,-,+] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 5
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 5
[-,+,+,-,+] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 3
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[+,-,-,+,+] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[+,-,+,-,+] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
[+,-,+,+,-] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
[+,+,-,-,+] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[+,+,-,+,-] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[+,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1
[-,-,-,+,+] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 5
[-,-,+,-,+] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 5
[-,-,+,+,-] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[-,+,-,-,+] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3
[-,+,-,+,-] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[-,+,+,-,-] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[+,-,-,-,+] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[+,-,-,+,-] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
[+,-,+,-,-] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[+,+,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1
[-,-,-,-,+] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1
[-,-,-,+,-] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1
[-,-,+,-,-] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[-,+,-,-,-] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[+,-,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1
[-,-,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[-,+,+,5,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 3
[+,-,+,5,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
[+,+,-,5,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[-,-,+,5,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 5
[-,+,-,5,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3
[+,-,-,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[-,-,-,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1
[+,+,4,3,+] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
[-,+,4,3,+] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 5
[+,-,4,3,+] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[+,+,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[-,-,4,3,+] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 5
[-,+,4,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[+,-,4,3,-] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
[-,-,4,3,-] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1
[+,+,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[-,+,4,5,3] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3
Description
The number of reduced words of a signed permutation.
This is the number of ways to write a permutation as a minimal length product of simple reflections.
Matching statistic: St001857
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[+] => [1] => [1] => 0 = 1 - 1
[-] => [1] => [1] => 0 = 1 - 1
[+,+] => [1,2] => [1,2] => 0 = 1 - 1
[-,+] => [2,1] => [2,1] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[+,-,+] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,3,2] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[2,1,+] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[2,1,-] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,+,1] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[3,-,1] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => ? = 1 - 1
[+,-,+,+] => [1,3,4,2] => [1,3,4,2] => ? = 1 - 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => ? = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => ? = 2 - 1
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => ? = 2 - 1
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => ? = 1 - 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => ? = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => ? = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => ? = 1 - 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => ? = 1 - 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ? = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => ? = 2 - 1
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => ? = 1 - 1
[-,-,4,3] => [3,1,2,4] => [3,1,2,4] => ? = 1 - 1
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => ? = 1 - 1
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => ? = 2 - 1
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => ? = 1 - 1
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => ? = 1 - 1
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[-,4,2,3] => [2,3,1,4] => [2,3,1,4] => ? = 2 - 1
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => ? = 1 - 1
[-,4,+,2] => [3,2,1,4] => [3,2,1,4] => ? = 2 - 1
[+,4,-,2] => [1,2,4,3] => [1,2,4,3] => ? = 1 - 1
[-,4,-,2] => [2,1,4,3] => [2,1,4,3] => ? = 1 - 1
[2,1,+,+] => [1,3,4,2] => [1,3,4,2] => ? = 1 - 1
[2,1,-,+] => [1,4,2,3] => [1,4,2,3] => ? = 2 - 1
[2,1,+,-] => [1,3,2,4] => [1,3,2,4] => ? = 1 - 1
[2,1,-,-] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[2,1,4,3] => [1,3,2,4] => [1,3,2,4] => ? = 2 - 1
[2,3,1,+] => [1,4,2,3] => [1,4,2,3] => ? = 2 - 1
[2,3,1,-] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => ? = 2 - 1
[2,4,+,1] => [3,1,2,4] => [3,1,2,4] => ? = 2 - 1
[2,4,-,1] => [1,2,4,3] => [1,2,4,3] => ? = 1 - 1
[3,1,2,+] => [1,2,4,3] => [1,2,4,3] => ? = 1 - 1
[3,1,2,-] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => ? = 2 - 1
[3,+,1,+] => [2,1,4,3] => [2,1,4,3] => ? = 1 - 1
[3,-,1,+] => [1,4,3,2] => [1,4,3,2] => ? = 2 - 1
[3,+,1,-] => [2,1,3,4] => [2,1,3,4] => ? = 1 - 1
[3,-,1,-] => [1,3,2,4] => [1,3,2,4] => ? = 1 - 1
Description
The number of edges in the reduced word graph of a signed permutation.
The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St000084
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[+] => [1] => [.,.]
=> [[],[]]
=> 2 = 1 + 1
[-] => [1] => [.,.]
=> [[],[]]
=> 2 = 1 + 1
[+,+] => [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[-,+] => [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[+,-] => [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[-,-] => [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[2,1] => [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[+,+,+] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[-,+,+] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[+,-,+] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[+,+,-] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[-,-,+] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[-,+,-] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[+,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[-,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[+,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 2 = 1 + 1
[-,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 2 = 1 + 1
[2,1,+] => [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[2,1,-] => [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[2,3,1] => [2,3,1] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 2 = 1 + 1
[3,+,1] => [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 2 = 1 + 1
[3,-,1] => [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 2 = 1 + 1
[+,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,-,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,+,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,-,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 2 + 1
[-,+,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 2 + 1
[-,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,-,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,-,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,-,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,-,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ? = 1 + 1
[-,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ? = 2 + 1
[+,-,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ? = 1 + 1
[-,-,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ? = 1 + 1
[+,3,2,+] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[-,3,2,+] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 2 + 1
[+,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[-,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[+,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[-,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[+,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> ? = 1 + 1
[-,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> ? = 2 + 1
[+,4,+,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ? = 1 + 1
[-,4,+,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ? = 2 + 1
[+,4,-,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ? = 1 + 1
[-,4,-,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ? = 1 + 1
[2,1,+,+] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,1,-,+] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 2 + 1
[2,1,+,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,1,-,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ? = 2 + 1
[2,3,1,+] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 2 + 1
[2,3,1,-] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,4,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ? = 2 + 1
[2,4,+,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ? = 2 + 1
[2,4,-,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ? = 1 + 1
[3,1,2,+] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 1
[3,1,2,-] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 1
[3,1,4,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 2 + 1
[3,+,1,+] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 1
[3,-,1,+] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 2 + 1
[3,+,1,-] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 1
[3,-,1,-] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 1
Description
The number of subtrees.
Matching statistic: St000328
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000328: Ordered trees ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000328: Ordered trees ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[+] => [1] => [.,.]
=> [[],[]]
=> 2 = 1 + 1
[-] => [1] => [.,.]
=> [[],[]]
=> 2 = 1 + 1
[+,+] => [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[-,+] => [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[+,-] => [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[-,-] => [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[2,1] => [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[+,+,+] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[-,+,+] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[+,-,+] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[+,+,-] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[-,-,+] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[-,+,-] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[+,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[-,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[+,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 2 = 1 + 1
[-,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 2 = 1 + 1
[2,1,+] => [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[2,1,-] => [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[2,3,1] => [2,3,1] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 2 = 1 + 1
[3,+,1] => [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 2 = 1 + 1
[3,-,1] => [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 2 = 1 + 1
[+,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,-,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,+,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,-,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 2 + 1
[-,+,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 2 + 1
[-,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,-,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,-,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,-,-,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,-,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[-,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ? = 1 + 1
[+,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ? = 1 + 1
[-,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ? = 2 + 1
[+,-,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ? = 1 + 1
[-,-,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ? = 1 + 1
[+,3,2,+] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[-,3,2,+] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 2 + 1
[+,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[-,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[+,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[-,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ? = 1 + 1
[+,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> ? = 1 + 1
[-,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> ? = 2 + 1
[+,4,+,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ? = 1 + 1
[-,4,+,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ? = 2 + 1
[+,4,-,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ? = 1 + 1
[-,4,-,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ? = 1 + 1
[2,1,+,+] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,1,-,+] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 2 + 1
[2,1,+,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,1,-,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ? = 2 + 1
[2,3,1,+] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 2 + 1
[2,3,1,-] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 1 + 1
[2,4,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ? = 2 + 1
[2,4,+,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ? = 2 + 1
[2,4,-,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ? = 1 + 1
[3,1,2,+] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 1
[3,1,2,-] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 1
[3,1,4,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 2 + 1
[3,+,1,+] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 1
[3,-,1,+] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 2 + 1
[3,+,1,-] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 1
[3,-,1,-] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 1
Description
The maximum number of child nodes in a tree.
Matching statistic: St001805
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St001805: Semistandard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St001805: Semistandard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[+] => [1] => [[1]]
=> [[1]]
=> ? = 1
[-] => [1] => [[1]]
=> [[1]]
=> ? = 1
[+,+] => [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1
[-,+] => [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1
[+,-] => [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1
[-,-] => [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1
[2,1] => [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[+,+,+] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 1
[-,+,+] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 1
[+,-,+] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 1
[+,+,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 1
[-,-,+] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 1
[-,+,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 1
[+,-,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 1
[-,-,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 1
[+,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> 1
[-,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> 1
[2,1,+] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> 1
[2,1,-] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> 1
[2,3,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> 1
[3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> 1
[3,+,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> 1
[3,-,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> 1
[+,+,+,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[-,+,+,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[+,-,+,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[+,+,-,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[+,+,+,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[-,-,+,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[-,+,-,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[-,+,+,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[+,-,-,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[+,-,+,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[+,+,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[-,-,-,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[-,-,+,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[-,+,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[+,-,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[-,-,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
[+,+,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ? = 1
[-,+,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ? = 2
[+,-,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ? = 1
[-,-,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ? = 1
[+,3,2,+] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ? = 1
[-,3,2,+] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ? = 2
[+,3,2,-] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ? = 1
[-,3,2,-] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ? = 1
[+,3,4,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ? = 1
[-,3,4,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ? = 1
[+,4,2,3] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ? = 1
[-,4,2,3] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ? = 2
[+,4,+,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ? = 1
[-,4,+,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ? = 2
[+,4,-,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ? = 1
[-,4,-,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ? = 1
[2,1,+,+] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 1
[2,1,-,+] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 2
[2,1,+,-] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 1
[2,1,-,-] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 1
[2,1,4,3] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ? = 2
[2,3,1,+] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ? = 2
[2,3,1,-] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ? = 1
[2,3,4,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ? = 1
[2,4,1,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ? = 2
[2,4,+,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ? = 2
[2,4,-,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ? = 1
[3,1,2,+] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ? = 1
[3,1,2,-] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ? = 1
[3,1,4,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ? = 2
[3,+,1,+] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ? = 1
[3,-,1,+] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ? = 2
Description
The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
A cylindrical tableau associated with a semistandard Young tableau $T$ is the skew semistandard tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St001630
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[+] => [1] => [1] => ([(0,1)],2)
=> ? = 1 + 1
[-] => [1] => [1] => ([(0,1)],2)
=> ? = 1 + 1
[+,+] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[-,+] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[+,-] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[-,-] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1] => [2,1] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[+,+,+] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[-,+,+] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[+,-,+] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[+,+,-] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[-,-,+] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[-,+,-] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[+,-,-] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[-,-,-] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[+,3,2] => [1,3,2] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[-,3,2] => [1,3,2] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,1,+] => [2,1,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,1,-] => [2,1,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,+,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,-,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 + 1
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 + 1
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,+,4,3] => [1,2,4,3] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,+,4,3] => [1,2,4,3] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 + 1
[+,-,4,3] => [1,2,4,3] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,-,4,3] => [1,2,4,3] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,3,2,+] => [1,3,2,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,3,2,+] => [1,3,2,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 + 1
[+,3,2,-] => [1,3,2,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,3,2,-] => [1,3,2,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,3,4,2] => [1,3,4,2] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[-,3,4,2] => [1,3,4,2] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[+,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
[-,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2 + 1
[+,4,+,2] => [1,4,3,2] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
[-,4,+,2] => [1,4,3,2] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2 + 1
[+,4,-,2] => [1,4,3,2] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
[-,4,-,2] => [1,4,3,2] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
[2,1,+,+] => [2,1,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[2,1,-,+] => [2,1,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 + 1
[2,1,+,-] => [2,1,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[2,1,-,-] => [2,1,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 + 1
[2,3,1,+] => [2,3,1,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 + 1
[2,3,1,-] => [2,3,1,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[2,4,1,3] => [2,4,1,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2 + 1
[2,4,+,1] => [2,4,3,1] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2 + 1
[2,4,-,1] => [2,4,3,1] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
[3,1,2,+] => [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,1,2,-] => [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,1,4,2] => [3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 2 + 1
[3,+,1,+] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,-,1,+] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 + 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
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