Your data matches 223 different statistics following compositions of up to 3 maps.
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Matching statistic: St000290
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 0 = -1 + 1
1 => [1] => 1 => 0 = -1 + 1
00 => [2] => 10 => 1 = 0 + 1
01 => [1,1] => 11 => 0 = -1 + 1
10 => [1,1] => 11 => 0 = -1 + 1
11 => [2] => 10 => 1 = 0 + 1
000 => [3] => 100 => 1 = 0 + 1
001 => [2,1] => 101 => 1 = 0 + 1
010 => [1,1,1] => 111 => 0 = -1 + 1
101 => [1,1,1] => 111 => 0 = -1 + 1
110 => [2,1] => 101 => 1 = 0 + 1
111 => [3] => 100 => 1 = 0 + 1
0000 => [4] => 1000 => 1 = 0 + 1
0001 => [3,1] => 1001 => 1 = 0 + 1
0010 => [2,1,1] => 1011 => 1 = 0 + 1
0101 => [1,1,1,1] => 1111 => 0 = -1 + 1
1010 => [1,1,1,1] => 1111 => 0 = -1 + 1
1101 => [2,1,1] => 1011 => 1 = 0 + 1
1110 => [3,1] => 1001 => 1 = 0 + 1
1111 => [4] => 1000 => 1 = 0 + 1
00000 => [5] => 10000 => 1 = 0 + 1
00001 => [4,1] => 10001 => 1 = 0 + 1
00010 => [3,1,1] => 10011 => 1 = 0 + 1
00101 => [2,1,1,1] => 10111 => 1 = 0 + 1
01010 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
10101 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
11010 => [2,1,1,1] => 10111 => 1 = 0 + 1
11101 => [3,1,1] => 10011 => 1 = 0 + 1
11110 => [4,1] => 10001 => 1 = 0 + 1
11111 => [5] => 10000 => 1 = 0 + 1
000000 => [6] => 100000 => 1 = 0 + 1
000001 => [5,1] => 100001 => 1 = 0 + 1
000010 => [4,1,1] => 100011 => 1 = 0 + 1
000101 => [3,1,1,1] => 100111 => 1 = 0 + 1
001010 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
010101 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
101010 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
110101 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
111010 => [3,1,1,1] => 100111 => 1 = 0 + 1
111101 => [4,1,1] => 100011 => 1 = 0 + 1
111110 => [5,1] => 100001 => 1 = 0 + 1
111111 => [6] => 100000 => 1 = 0 + 1
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000291
(load all 6 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 0 = -1 + 1
1 => [1] => 1 => 0 = -1 + 1
00 => [2] => 10 => 1 = 0 + 1
01 => [1,1] => 11 => 0 = -1 + 1
10 => [1,1] => 11 => 0 = -1 + 1
11 => [2] => 10 => 1 = 0 + 1
000 => [3] => 100 => 1 = 0 + 1
001 => [2,1] => 101 => 1 = 0 + 1
010 => [1,1,1] => 111 => 0 = -1 + 1
101 => [1,1,1] => 111 => 0 = -1 + 1
110 => [2,1] => 101 => 1 = 0 + 1
111 => [3] => 100 => 1 = 0 + 1
0000 => [4] => 1000 => 1 = 0 + 1
0001 => [3,1] => 1001 => 1 = 0 + 1
0010 => [2,1,1] => 1011 => 1 = 0 + 1
0101 => [1,1,1,1] => 1111 => 0 = -1 + 1
1010 => [1,1,1,1] => 1111 => 0 = -1 + 1
1101 => [2,1,1] => 1011 => 1 = 0 + 1
1110 => [3,1] => 1001 => 1 = 0 + 1
1111 => [4] => 1000 => 1 = 0 + 1
00000 => [5] => 10000 => 1 = 0 + 1
00001 => [4,1] => 10001 => 1 = 0 + 1
00010 => [3,1,1] => 10011 => 1 = 0 + 1
00101 => [2,1,1,1] => 10111 => 1 = 0 + 1
01010 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
10101 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
11010 => [2,1,1,1] => 10111 => 1 = 0 + 1
11101 => [3,1,1] => 10011 => 1 = 0 + 1
11110 => [4,1] => 10001 => 1 = 0 + 1
11111 => [5] => 10000 => 1 = 0 + 1
000000 => [6] => 100000 => 1 = 0 + 1
000001 => [5,1] => 100001 => 1 = 0 + 1
000010 => [4,1,1] => 100011 => 1 = 0 + 1
000101 => [3,1,1,1] => 100111 => 1 = 0 + 1
001010 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
010101 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
101010 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
110101 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
111010 => [3,1,1,1] => 100111 => 1 = 0 + 1
111101 => [4,1,1] => 100011 => 1 = 0 + 1
111110 => [5,1] => 100001 => 1 = 0 + 1
111111 => [6] => 100000 => 1 = 0 + 1
Description
The number of descents of a binary word.
Matching statistic: St000480
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
 => 0 = -1 + 1
1 => [1] => [1]
 => 0 = -1 + 1
00 => [2] => [2]
 => 1 = 0 + 1
01 => [1,1] => [1,1]
 => 0 = -1 + 1
10 => [1,1] => [1,1]
 => 0 = -1 + 1
11 => [2] => [2]
 => 1 = 0 + 1
000 => [3] => [3]
 => 1 = 0 + 1
001 => [2,1] => [2,1]
 => 1 = 0 + 1
010 => [1,1,1] => [1,1,1]
 => 0 = -1 + 1
101 => [1,1,1] => [1,1,1]
 => 0 = -1 + 1
110 => [2,1] => [2,1]
 => 1 = 0 + 1
111 => [3] => [3]
 => 1 = 0 + 1
0000 => [4] => [4]
 => 1 = 0 + 1
0001 => [3,1] => [3,1]
 => 1 = 0 + 1
0010 => [2,1,1] => [2,1,1]
 => 1 = 0 + 1
0101 => [1,1,1,1] => [1,1,1,1]
 => 0 = -1 + 1
1010 => [1,1,1,1] => [1,1,1,1]
 => 0 = -1 + 1
1101 => [2,1,1] => [2,1,1]
 => 1 = 0 + 1
1110 => [3,1] => [3,1]
 => 1 = 0 + 1
1111 => [4] => [4]
 => 1 = 0 + 1
00000 => [5] => [5]
 => 1 = 0 + 1
00001 => [4,1] => [4,1]
 => 1 = 0 + 1
00010 => [3,1,1] => [3,1,1]
 => 1 = 0 + 1
00101 => [2,1,1,1] => [2,1,1,1]
 => 1 = 0 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
 => 0 = -1 + 1
10101 => [1,1,1,1,1] => [1,1,1,1,1]
 => 0 = -1 + 1
11010 => [2,1,1,1] => [2,1,1,1]
 => 1 = 0 + 1
11101 => [3,1,1] => [3,1,1]
 => 1 = 0 + 1
11110 => [4,1] => [4,1]
 => 1 = 0 + 1
11111 => [5] => [5]
 => 1 = 0 + 1
000000 => [6] => [6]
 => 1 = 0 + 1
000001 => [5,1] => [5,1]
 => 1 = 0 + 1
000010 => [4,1,1] => [4,1,1]
 => 1 = 0 + 1
000101 => [3,1,1,1] => [3,1,1,1]
 => 1 = 0 + 1
001010 => [2,1,1,1,1] => [2,1,1,1,1]
 => 1 = 0 + 1
010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 0 = -1 + 1
101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 0 = -1 + 1
110101 => [2,1,1,1,1] => [2,1,1,1,1]
 => 1 = 0 + 1
111010 => [3,1,1,1] => [3,1,1,1]
 => 1 = 0 + 1
111101 => [4,1,1] => [4,1,1]
 => 1 = 0 + 1
111110 => [5,1] => [5,1]
 => 1 = 0 + 1
111111 => [6] => [6]
 => 1 = 0 + 1
Description
The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000628
(load all 7 compositions to match this statistic)
Mapping
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = -1 + 1
1 => 1 => 1 => 0 = -1 + 1
00 => 01 => 10 => 1 = 0 + 1
01 => 00 => 00 => 0 = -1 + 1
10 => 11 => 11 => 0 = -1 + 1
11 => 10 => 01 => 1 = 0 + 1
000 => 010 => 001 => 1 = 0 + 1
001 => 011 => 101 => 1 = 0 + 1
010 => 000 => 000 => 0 = -1 + 1
101 => 111 => 111 => 0 = -1 + 1
110 => 100 => 010 => 1 = 0 + 1
111 => 101 => 110 => 1 = 0 + 1
0000 => 0101 => 1010 => 1 = 0 + 1
0001 => 0100 => 0010 => 1 = 0 + 1
0010 => 0111 => 1011 => 1 = 0 + 1
0101 => 0000 => 0000 => 0 = -1 + 1
1010 => 1111 => 1111 => 0 = -1 + 1
1101 => 1000 => 0100 => 1 = 0 + 1
1110 => 1011 => 1101 => 1 = 0 + 1
1111 => 1010 => 0101 => 1 = 0 + 1
00000 => 01010 => 00101 => 1 = 0 + 1
00001 => 01011 => 10101 => 1 = 0 + 1
00010 => 01000 => 00100 => 1 = 0 + 1
00101 => 01111 => 10111 => 1 = 0 + 1
01010 => 00000 => 00000 => 0 = -1 + 1
10101 => 11111 => 11111 => 0 = -1 + 1
11010 => 10000 => 01000 => 1 = 0 + 1
11101 => 10111 => 11011 => 1 = 0 + 1
11110 => 10100 => 01010 => 1 = 0 + 1
11111 => 10101 => 11010 => 1 = 0 + 1
000000 => 010101 => 101010 => 1 = 0 + 1
000001 => 010100 => 001010 => 1 = 0 + 1
000010 => 010111 => 101011 => 1 = 0 + 1
000101 => 010000 => 001000 => 1 = 0 + 1
001010 => 011111 => 101111 => 1 = 0 + 1
010101 => 000000 => 000000 => 0 = -1 + 1
101010 => 111111 => 111111 => 0 = -1 + 1
110101 => 100000 => 010000 => 1 = 0 + 1
111010 => 101111 => 110111 => 1 = 0 + 1
111101 => 101000 => 010100 => 1 = 0 + 1
111110 => 101011 => 110101 => 1 = 0 + 1
111111 => 101010 => 010101 => 1 = 0 + 1
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St000875
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 0 = -1 + 1
1 => [1] => 1 => 0 = -1 + 1
00 => [2] => 10 => 1 = 0 + 1
01 => [1,1] => 11 => 0 = -1 + 1
10 => [1,1] => 11 => 0 = -1 + 1
11 => [2] => 10 => 1 = 0 + 1
000 => [3] => 100 => 1 = 0 + 1
001 => [2,1] => 101 => 1 = 0 + 1
010 => [1,1,1] => 111 => 0 = -1 + 1
101 => [1,1,1] => 111 => 0 = -1 + 1
110 => [2,1] => 101 => 1 = 0 + 1
111 => [3] => 100 => 1 = 0 + 1
0000 => [4] => 1000 => 1 = 0 + 1
0001 => [3,1] => 1001 => 1 = 0 + 1
0010 => [2,1,1] => 1011 => 1 = 0 + 1
0101 => [1,1,1,1] => 1111 => 0 = -1 + 1
1010 => [1,1,1,1] => 1111 => 0 = -1 + 1
1101 => [2,1,1] => 1011 => 1 = 0 + 1
1110 => [3,1] => 1001 => 1 = 0 + 1
1111 => [4] => 1000 => 1 = 0 + 1
00000 => [5] => 10000 => 1 = 0 + 1
00001 => [4,1] => 10001 => 1 = 0 + 1
00010 => [3,1,1] => 10011 => 1 = 0 + 1
00101 => [2,1,1,1] => 10111 => 1 = 0 + 1
01010 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
10101 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
11010 => [2,1,1,1] => 10111 => 1 = 0 + 1
11101 => [3,1,1] => 10011 => 1 = 0 + 1
11110 => [4,1] => 10001 => 1 = 0 + 1
11111 => [5] => 10000 => 1 = 0 + 1
000000 => [6] => 100000 => 1 = 0 + 1
000001 => [5,1] => 100001 => 1 = 0 + 1
000010 => [4,1,1] => 100011 => 1 = 0 + 1
000101 => [3,1,1,1] => 100111 => 1 = 0 + 1
001010 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
010101 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
101010 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
110101 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
111010 => [3,1,1,1] => 100111 => 1 = 0 + 1
111101 => [4,1,1] => 100011 => 1 = 0 + 1
111110 => [5,1] => 100001 => 1 = 0 + 1
111111 => [6] => 100000 => 1 = 0 + 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001280
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
 => 0 = -1 + 1
1 => [1] => [1]
 => 0 = -1 + 1
00 => [2] => [2]
 => 1 = 0 + 1
01 => [1,1] => [1,1]
 => 0 = -1 + 1
10 => [1,1] => [1,1]
 => 0 = -1 + 1
11 => [2] => [2]
 => 1 = 0 + 1
000 => [3] => [3]
 => 1 = 0 + 1
001 => [2,1] => [2,1]
 => 1 = 0 + 1
010 => [1,1,1] => [1,1,1]
 => 0 = -1 + 1
101 => [1,1,1] => [1,1,1]
 => 0 = -1 + 1
110 => [2,1] => [2,1]
 => 1 = 0 + 1
111 => [3] => [3]
 => 1 = 0 + 1
0000 => [4] => [4]
 => 1 = 0 + 1
0001 => [3,1] => [3,1]
 => 1 = 0 + 1
0010 => [2,1,1] => [2,1,1]
 => 1 = 0 + 1
0101 => [1,1,1,1] => [1,1,1,1]
 => 0 = -1 + 1
1010 => [1,1,1,1] => [1,1,1,1]
 => 0 = -1 + 1
1101 => [2,1,1] => [2,1,1]
 => 1 = 0 + 1
1110 => [3,1] => [3,1]
 => 1 = 0 + 1
1111 => [4] => [4]
 => 1 = 0 + 1
00000 => [5] => [5]
 => 1 = 0 + 1
00001 => [4,1] => [4,1]
 => 1 = 0 + 1
00010 => [3,1,1] => [3,1,1]
 => 1 = 0 + 1
00101 => [2,1,1,1] => [2,1,1,1]
 => 1 = 0 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
 => 0 = -1 + 1
10101 => [1,1,1,1,1] => [1,1,1,1,1]
 => 0 = -1 + 1
11010 => [2,1,1,1] => [2,1,1,1]
 => 1 = 0 + 1
11101 => [3,1,1] => [3,1,1]
 => 1 = 0 + 1
11110 => [4,1] => [4,1]
 => 1 = 0 + 1
11111 => [5] => [5]
 => 1 = 0 + 1
000000 => [6] => [6]
 => 1 = 0 + 1
000001 => [5,1] => [5,1]
 => 1 = 0 + 1
000010 => [4,1,1] => [4,1,1]
 => 1 = 0 + 1
000101 => [3,1,1,1] => [3,1,1,1]
 => 1 = 0 + 1
001010 => [2,1,1,1,1] => [2,1,1,1,1]
 => 1 = 0 + 1
010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 0 = -1 + 1
101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 0 = -1 + 1
110101 => [2,1,1,1,1] => [2,1,1,1,1]
 => 1 = 0 + 1
111010 => [3,1,1,1] => [3,1,1,1]
 => 1 = 0 + 1
111101 => [4,1,1] => [4,1,1]
 => 1 = 0 + 1
111110 => [5,1] => [5,1]
 => 1 = 0 + 1
111111 => [6] => [6]
 => 1 = 0 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001421
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001421: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 0 = -1 + 1
1 => [1] => 1 => 0 = -1 + 1
00 => [2] => 10 => 1 = 0 + 1
01 => [1,1] => 11 => 0 = -1 + 1
10 => [1,1] => 11 => 0 = -1 + 1
11 => [2] => 10 => 1 = 0 + 1
000 => [3] => 100 => 1 = 0 + 1
001 => [2,1] => 101 => 1 = 0 + 1
010 => [1,1,1] => 111 => 0 = -1 + 1
101 => [1,1,1] => 111 => 0 = -1 + 1
110 => [2,1] => 101 => 1 = 0 + 1
111 => [3] => 100 => 1 = 0 + 1
0000 => [4] => 1000 => 1 = 0 + 1
0001 => [3,1] => 1001 => 1 = 0 + 1
0010 => [2,1,1] => 1011 => 1 = 0 + 1
0101 => [1,1,1,1] => 1111 => 0 = -1 + 1
1010 => [1,1,1,1] => 1111 => 0 = -1 + 1
1101 => [2,1,1] => 1011 => 1 = 0 + 1
1110 => [3,1] => 1001 => 1 = 0 + 1
1111 => [4] => 1000 => 1 = 0 + 1
00000 => [5] => 10000 => 1 = 0 + 1
00001 => [4,1] => 10001 => 1 = 0 + 1
00010 => [3,1,1] => 10011 => 1 = 0 + 1
00101 => [2,1,1,1] => 10111 => 1 = 0 + 1
01010 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
10101 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
11010 => [2,1,1,1] => 10111 => 1 = 0 + 1
11101 => [3,1,1] => 10011 => 1 = 0 + 1
11110 => [4,1] => 10001 => 1 = 0 + 1
11111 => [5] => 10000 => 1 = 0 + 1
000000 => [6] => 100000 => 1 = 0 + 1
000001 => [5,1] => 100001 => 1 = 0 + 1
000010 => [4,1,1] => 100011 => 1 = 0 + 1
000101 => [3,1,1,1] => 100111 => 1 = 0 + 1
001010 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
010101 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
101010 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
110101 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
111010 => [3,1,1,1] => 100111 => 1 = 0 + 1
111101 => [4,1,1] => 100011 => 1 = 0 + 1
111110 => [5,1] => 100001 => 1 = 0 + 1
111111 => [6] => 100000 => 1 = 0 + 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St001485
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001485: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 0 = -1 + 1
1 => [1] => 1 => 0 = -1 + 1
00 => [2] => 10 => 1 = 0 + 1
01 => [1,1] => 11 => 0 = -1 + 1
10 => [1,1] => 11 => 0 = -1 + 1
11 => [2] => 10 => 1 = 0 + 1
000 => [3] => 100 => 1 = 0 + 1
001 => [2,1] => 101 => 1 = 0 + 1
010 => [1,1,1] => 111 => 0 = -1 + 1
101 => [1,1,1] => 111 => 0 = -1 + 1
110 => [2,1] => 101 => 1 = 0 + 1
111 => [3] => 100 => 1 = 0 + 1
0000 => [4] => 1000 => 1 = 0 + 1
0001 => [3,1] => 1001 => 1 = 0 + 1
0010 => [2,1,1] => 1011 => 1 = 0 + 1
0101 => [1,1,1,1] => 1111 => 0 = -1 + 1
1010 => [1,1,1,1] => 1111 => 0 = -1 + 1
1101 => [2,1,1] => 1011 => 1 = 0 + 1
1110 => [3,1] => 1001 => 1 = 0 + 1
1111 => [4] => 1000 => 1 = 0 + 1
00000 => [5] => 10000 => 1 = 0 + 1
00001 => [4,1] => 10001 => 1 = 0 + 1
00010 => [3,1,1] => 10011 => 1 = 0 + 1
00101 => [2,1,1,1] => 10111 => 1 = 0 + 1
01010 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
10101 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
11010 => [2,1,1,1] => 10111 => 1 = 0 + 1
11101 => [3,1,1] => 10011 => 1 = 0 + 1
11110 => [4,1] => 10001 => 1 = 0 + 1
11111 => [5] => 10000 => 1 = 0 + 1
000000 => [6] => 100000 => 1 = 0 + 1
000001 => [5,1] => 100001 => 1 = 0 + 1
000010 => [4,1,1] => 100011 => 1 = 0 + 1
000101 => [3,1,1,1] => 100111 => 1 = 0 + 1
001010 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
010101 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
101010 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
110101 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
111010 => [3,1,1,1] => 100111 => 1 = 0 + 1
111101 => [4,1,1] => 100011 => 1 = 0 + 1
111110 => [5,1] => 100001 => 1 = 0 + 1
111111 => [6] => 100000 => 1 = 0 + 1
Description
The modular major index of a binary word. This is [[St000290]] modulo the length of the word.
Matching statistic: St000630
(load all 12 compositions to match this statistic)
Mapping
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 1 = -1 + 2
1 => 1 => 1 => 1 = -1 + 2
00 => 01 => 10 => 2 = 0 + 2
01 => 00 => 00 => 1 = -1 + 2
10 => 11 => 11 => 1 = -1 + 2
11 => 10 => 01 => 2 = 0 + 2
000 => 010 => 100 => 2 = 0 + 2
001 => 011 => 110 => 2 = 0 + 2
010 => 000 => 000 => 1 = -1 + 2
101 => 111 => 111 => 1 = -1 + 2
110 => 100 => 001 => 2 = 0 + 2
111 => 101 => 011 => 2 = 0 + 2
0000 => 0101 => 1010 => 2 = 0 + 2
0001 => 0100 => 1000 => 2 = 0 + 2
0010 => 0111 => 1110 => 2 = 0 + 2
0101 => 0000 => 0000 => 1 = -1 + 2
1010 => 1111 => 1111 => 1 = -1 + 2
1101 => 1000 => 0001 => 2 = 0 + 2
1110 => 1011 => 0111 => 2 = 0 + 2
1111 => 1010 => 0101 => 2 = 0 + 2
00000 => 01010 => 10100 => 2 = 0 + 2
00001 => 01011 => 10110 => 2 = 0 + 2
00010 => 01000 => 10000 => 2 = 0 + 2
00101 => 01111 => 11110 => 2 = 0 + 2
01010 => 00000 => 00000 => 1 = -1 + 2
10101 => 11111 => 11111 => 1 = -1 + 2
11010 => 10000 => 00001 => 2 = 0 + 2
11101 => 10111 => 01111 => 2 = 0 + 2
11110 => 10100 => 01001 => 2 = 0 + 2
11111 => 10101 => 01011 => 2 = 0 + 2
000000 => 010101 => 101010 => 2 = 0 + 2
000001 => 010100 => 101000 => 2 = 0 + 2
000010 => 010111 => 101110 => 2 = 0 + 2
000101 => 010000 => 100000 => 2 = 0 + 2
001010 => 011111 => 111110 => 2 = 0 + 2
010101 => 000000 => 000000 => 1 = -1 + 2
101010 => 111111 => 111111 => 1 = -1 + 2
110101 => 100000 => 000001 => 2 = 0 + 2
111010 => 101111 => 011111 => 2 = 0 + 2
111101 => 101000 => 010001 => 2 = 0 + 2
111110 => 101011 => 010111 => 2 = 0 + 2
111111 => 101010 => 010101 => 2 = 0 + 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: St001642
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001642: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
 => 1 = -1 + 2
1 => [1] => ([],1)
 => 1 = -1 + 2
00 => [2] => ([],2)
 => 2 = 0 + 2
01 => [1,1] => ([(0,1)],2)
 => 1 = -1 + 2
10 => [1,1] => ([(0,1)],2)
 => 1 = -1 + 2
11 => [2] => ([],2)
 => 2 = 0 + 2
000 => [3] => ([],3)
 => 2 = 0 + 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = -1 + 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = -1 + 2
110 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
111 => [3] => ([],3)
 => 2 = 0 + 2
0000 => [4] => ([],4)
 => 2 = 0 + 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = -1 + 2
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = -1 + 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
1111 => [4] => ([],4)
 => 2 = 0 + 2
00000 => [5] => ([],5)
 => 2 = 0 + 2
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 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)
 => 2 = 0 + 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)
 => 1 = -1 + 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)
 => 1 = -1 + 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)
 => 2 = 0 + 2
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
11111 => [5] => ([],5)
 => 2 = 0 + 2
000000 => [6] => ([],6)
 => 2 = 0 + 2
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 0 + 2
000101 => [3,1,1,1] => ([(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)
 => 2 = 0 + 2
001010 => [2,1,1,1,1] => ([(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)
 => 2 = 0 + 2
010101 => [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 = -1 + 2
101010 => [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 = -1 + 2
110101 => [2,1,1,1,1] => ([(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)
 => 2 = 0 + 2
111010 => [3,1,1,1] => ([(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)
 => 2 = 0 + 2
111101 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 0 + 2
111110 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
111111 => [6] => ([],6)
 => 2 = 0 + 2
Description
The Prague dimension of a graph. This is the least number of complete graphs such that the graph is an induced subgraph of their (categorical) product. Put differently, this is the least number $n$ such that the graph can be embedded into $\mathbb N^n$, where two points are connected by an edge if and only if they differ in all coordinates.
The following 213 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000455The second largest eigenvalue of a graph if it is integral. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000347The inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000390The number of runs of ones in a binary word. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000651The maximal size of a rise in a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000691The number of changes of a binary word. St000761The number of ascents in an integer composition. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001375The pancake length of a permutation. St001393The induced matching number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000058The order of a permutation. St000259The diameter of a connected graph. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000388The number of orbits of vertices of a graph under automorphisms. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000903The number of different parts of an integer composition. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000659The number of rises of length at least 2 of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000354The number of recoils of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000653The last descent of a permutation. St000730The maximal arc length of a set partition. St000794The mak of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000919The number of maximal left branches of a binary tree. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001114The number of odd descents of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000485The length of the longest cycle of a permutation. St000990The first ascent of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001352The number of internal nodes in the modular decomposition of a graph. St000260The radius of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000699The toughness times the least common multiple of 1,. St000023The number of inner peaks of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001592The maximal number of simple paths between any two different vertices of a graph. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000527The width of the poset. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001500The global dimension of magnitude 1 Nakayama algebras. St001734The lettericity of a graph. St000264The girth of a graph, which is not a tree. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001118The acyclic chromatic index of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000098The chromatic number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000806The semiperimeter of the associated bargraph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000307The number of rowmotion orbits of a poset. St000544The cop number of a graph. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St001746The coalition number of a graph. St000640The rank of the largest boolean interval in a poset. St000454The largest eigenvalue of a graph if it is integral. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.