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Your data matches 26 different statistics following compositions of up to 3 maps.
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Matching statistic: St000630
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(load all 8 compositions to match this statistic)
St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 1
01 => 2
10 => 2
11 => 1
000 => 1
001 => 2
010 => 1
011 => 2
100 => 2
101 => 1
110 => 2
111 => 1
0000 => 1
0001 => 2
0010 => 2
0011 => 2
0100 => 2
0101 => 2
0110 => 1
0111 => 2
1000 => 2
1001 => 1
1010 => 2
1011 => 2
1100 => 2
1101 => 2
1110 => 2
1111 => 1
00000 => 1
00001 => 2
00010 => 2
00011 => 2
00100 => 1
00101 => 2
00110 => 2
00111 => 2
01000 => 2
01001 => 2
01010 => 1
01011 => 2
01100 => 2
01101 => 2
01110 => 1
01111 => 2
10000 => 2
10001 => 1
10010 => 2
10011 => 2
Description
The length of the shortest palindromic decomposition of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
Matching statistic: St000402
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 1
1 => [1,1] => [1,0,1,0]
=> [1,2] => 1
00 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 2
Description
Half the size of the symmetry class of a permutation.
The symmetry class of a permutation $\pi$ is the set of all permutations that can be obtained from $\pi$ by the three elementary operations '''inverse''' ([[Mp00066]]), '''reverse''' ([[Mp00064]]), and '''complement''' ([[Mp00069]]).
This statistic is undefined for the unique permutation on one element, because its value would be $1/2$.
Matching statistic: St000526
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000526: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000526: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> ([(0,1)],2)
=> 1
1 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> 1
00 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
01 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2
10 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
11 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
000 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
001 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
010 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
100 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1
110 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
0000 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0001 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
0010 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
0100 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
1000 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
1001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
1100 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
00001 => [5,1] => [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
00010 => [4,2] => [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
00100 => [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 2
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 2
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
01000 => [2,4] => [[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 2
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
10000 => [1,5] => [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
10001 => [1,4,1] => [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 1
10010 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 2
Description
The number of posets with combinatorially isomorphic order polytopes.
Matching statistic: St001568
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => [1]
=> ? ∊ {1,1}
1 => 1 => [1] => [1]
=> ? ∊ {1,1}
00 => 01 => [1,1] => [1,1]
=> 2
01 => 10 => [1,1] => [1,1]
=> 2
10 => 11 => [2] => [2]
=> 1
11 => 11 => [2] => [2]
=> 1
000 => 001 => [2,1] => [2,1]
=> 1
001 => 010 => [1,1,1] => [1,1,1]
=> 2
010 => 101 => [1,1,1] => [1,1,1]
=> 2
011 => 101 => [1,1,1] => [1,1,1]
=> 2
100 => 101 => [1,1,1] => [1,1,1]
=> 2
101 => 110 => [2,1] => [2,1]
=> 1
110 => 111 => [3] => [3]
=> 1
111 => 111 => [3] => [3]
=> 1
0000 => 0001 => [3,1] => [3,1]
=> 1
0001 => 0010 => [2,1,1] => [2,1,1]
=> 2
0010 => 0101 => [1,1,1,1] => [1,1,1,1]
=> 2
0011 => 0101 => [1,1,1,1] => [1,1,1,1]
=> 2
0100 => 1001 => [1,2,1] => [2,1,1]
=> 2
0101 => 1010 => [1,1,1,1] => [1,1,1,1]
=> 2
0110 => 1011 => [1,1,2] => [2,1,1]
=> 2
0111 => 1011 => [1,1,2] => [2,1,1]
=> 2
1000 => 1001 => [1,2,1] => [2,1,1]
=> 2
1001 => 1010 => [1,1,1,1] => [1,1,1,1]
=> 2
1010 => 1101 => [2,1,1] => [2,1,1]
=> 2
1011 => 1101 => [2,1,1] => [2,1,1]
=> 2
1100 => 1101 => [2,1,1] => [2,1,1]
=> 2
1101 => 1110 => [3,1] => [3,1]
=> 1
1110 => 1111 => [4] => [4]
=> 1
1111 => 1111 => [4] => [4]
=> 1
00000 => 00001 => [4,1] => [4,1]
=> 1
00001 => 00010 => [3,1,1] => [3,1,1]
=> 2
00010 => 00101 => [2,1,1,1] => [2,1,1,1]
=> 2
00011 => 00101 => [2,1,1,1] => [2,1,1,1]
=> 2
00100 => 01001 => [1,1,2,1] => [2,1,1,1]
=> 2
00101 => 01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> 2
00110 => 01011 => [1,1,1,2] => [2,1,1,1]
=> 2
00111 => 01011 => [1,1,1,2] => [2,1,1,1]
=> 2
01000 => 10001 => [1,3,1] => [3,1,1]
=> 2
01001 => 10010 => [1,2,1,1] => [2,1,1,1]
=> 2
01010 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 2
01011 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 2
01100 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 2
01101 => 10110 => [1,1,2,1] => [2,1,1,1]
=> 2
01110 => 10111 => [1,1,3] => [3,1,1]
=> 2
01111 => 10111 => [1,1,3] => [3,1,1]
=> 2
10000 => 10001 => [1,3,1] => [3,1,1]
=> 2
10001 => 10010 => [1,2,1,1] => [2,1,1,1]
=> 2
10010 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 2
10011 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 2
10100 => 11001 => [2,2,1] => [2,2,1]
=> 1
10101 => 11010 => [2,1,1,1] => [2,1,1,1]
=> 2
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001195
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? ∊ {1,1} - 1
1 => [1] => [1,0]
=> [1,0]
=> ? ∊ {1,1} - 1
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> ? ∊ {1,1,2,2} - 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,1,2,2} - 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,1,2,2} - 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> ? ∊ {1,1,2,2} - 1
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St000259
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => ([(0,1)],2)
=> 1
1 => 1 => [1,1] => ([(0,1)],2)
=> 1
00 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
01 => 10 => [1,2] => ([(1,2)],3)
=> ? = 2
10 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
000 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
001 => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1}
010 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
011 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
100 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
101 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1}
110 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
0000 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
0001 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2}
0010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
0011 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
0100 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2}
0110 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
0111 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1000 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1001 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2}
1010 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1011 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1100 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1101 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2}
1110 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
1111 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
00000 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
00001 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2}
00010 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
00011 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
00100 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
00101 => 01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2}
00110 => 01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
00111 => 01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
01000 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
01001 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2}
01010 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
01011 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
01100 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
01101 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2}
01110 => 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
01111 => 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
10000 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
10001 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2}
10010 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
10011 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
10100 => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
10101 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2}
10110 => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
10111 => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
11000 => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
11001 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2}
11010 => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
11011 => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
11100 => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
11101 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2}
11110 => 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
11111 => 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => ([],1)
=> 0 = 1 - 1
1 => [1] => [1] => ([],1)
=> 0 = 1 - 1
00 => [2] => [1] => ([],1)
=> 0 = 1 - 1
01 => [1,1] => [2] => ([],2)
=> ? ∊ {2,2} - 1
10 => [1,1] => [2] => ([],2)
=> ? ∊ {2,2} - 1
11 => [2] => [1] => ([],1)
=> 0 = 1 - 1
000 => [3] => [1] => ([],1)
=> 0 = 1 - 1
001 => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
010 => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1} - 1
011 => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
100 => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
101 => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1} - 1
110 => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
111 => [3] => [1] => ([],1)
=> 0 = 1 - 1
0000 => [4] => [1] => ([],1)
=> 0 = 1 - 1
0001 => [3,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
0010 => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,2} - 1
0011 => [2,2] => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,2} - 1
0100 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
0101 => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,2,2,2,2} - 1
0110 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
0111 => [1,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
1000 => [1,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
1001 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
1010 => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,2,2,2,2} - 1
1011 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
1100 => [2,2] => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,2} - 1
1101 => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,2} - 1
1110 => [3,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
1111 => [4] => [1] => ([],1)
=> 0 = 1 - 1
00000 => [5] => [1] => ([],1)
=> 0 = 1 - 1
00001 => [4,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
00010 => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
00011 => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
00100 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
00101 => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
00110 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
00111 => [2,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
01000 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
01001 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
01010 => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
01011 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
01100 => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
01101 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
01110 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
01111 => [1,4] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
10000 => [1,4] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
10001 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
10010 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
10011 => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
10100 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
10101 => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
10110 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
10111 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
11000 => [2,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
11001 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
11010 => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
11011 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
11100 => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
11101 => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2} - 1
11110 => [4,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
11111 => [5] => [1] => ([],1)
=> 0 = 1 - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001043
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
1 => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
01 => [2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
000 => [4] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 2
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> 2
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> 1
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> 2
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> 2
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 2
Description
The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001630
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1}
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1}
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {2,2}
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {2,2}
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? ∊ {2,2}
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? ∊ {2,2}
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 1
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> ([(0,1)],2)
=> -1 = 1 - 2
1 => [1] => ([],1)
=> ([(0,1)],2)
=> -1 = 1 - 2
00 => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
01 => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
10 => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
11 => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
000 => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
001 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
011 => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1} - 2
100 => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1} - 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
110 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
111 => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
0000 => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
0011 => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2} - 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2} - 2
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 1 - 2
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2} - 2
0111 => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2} - 2
1000 => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2} - 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2} - 2
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 1 - 2
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2} - 2
1100 => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2} - 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
1111 => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
00000 => [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
00111 => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 1 - 2
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
01111 => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
10000 => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 1 - 2
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
11000 => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 2
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
11111 => [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001399The distinguishing number of a poset. St000544The cop number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001624The breadth of a lattice. St001644The dimension of a graph. St001330The hat guessing number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000741The Colin de Verdière graph invariant. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001118The acyclic chromatic index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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