Your data matches 83 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000392
(load all 3 compositions to match this statistic)
Mapping
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 1
00 => 0
01 => 1
10 => 1
11 => 2
000 => 0
001 => 1
010 => 1
011 => 2
100 => 1
101 => 1
110 => 2
111 => 3
0000 => 0
0001 => 1
0010 => 1
0011 => 2
0100 => 1
0101 => 1
0110 => 2
0111 => 3
1000 => 1
1001 => 1
1010 => 1
1011 => 2
1100 => 2
1101 => 2
1110 => 3
1111 => 4
00000 => 0
00001 => 1
00010 => 1
00011 => 2
00100 => 1
00101 => 1
00110 => 2
00111 => 3
01000 => 1
01001 => 1
01010 => 1
01011 => 2
01100 => 2
01101 => 2
01110 => 3
01111 => 4
10000 => 1
10001 => 1
10010 => 1
10011 => 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000381
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 2 = 1 + 1
1 => [1,1] => 1 = 0 + 1
00 => [3] => 3 = 2 + 1
01 => [2,1] => 2 = 1 + 1
10 => [1,2] => 2 = 1 + 1
11 => [1,1,1] => 1 = 0 + 1
000 => [4] => 4 = 3 + 1
001 => [3,1] => 3 = 2 + 1
010 => [2,2] => 2 = 1 + 1
011 => [2,1,1] => 2 = 1 + 1
100 => [1,3] => 3 = 2 + 1
101 => [1,2,1] => 2 = 1 + 1
110 => [1,1,2] => 2 = 1 + 1
111 => [1,1,1,1] => 1 = 0 + 1
0000 => [5] => 5 = 4 + 1
0001 => [4,1] => 4 = 3 + 1
0010 => [3,2] => 3 = 2 + 1
0011 => [3,1,1] => 3 = 2 + 1
0100 => [2,3] => 3 = 2 + 1
0101 => [2,2,1] => 2 = 1 + 1
0110 => [2,1,2] => 2 = 1 + 1
0111 => [2,1,1,1] => 2 = 1 + 1
1000 => [1,4] => 4 = 3 + 1
1001 => [1,3,1] => 3 = 2 + 1
1010 => [1,2,2] => 2 = 1 + 1
1011 => [1,2,1,1] => 2 = 1 + 1
1100 => [1,1,3] => 3 = 2 + 1
1101 => [1,1,2,1] => 2 = 1 + 1
1110 => [1,1,1,2] => 2 = 1 + 1
1111 => [1,1,1,1,1] => 1 = 0 + 1
00000 => [6] => 6 = 5 + 1
00001 => [5,1] => 5 = 4 + 1
00010 => [4,2] => 4 = 3 + 1
00011 => [4,1,1] => 4 = 3 + 1
00100 => [3,3] => 3 = 2 + 1
00101 => [3,2,1] => 3 = 2 + 1
00110 => [3,1,2] => 3 = 2 + 1
00111 => [3,1,1,1] => 3 = 2 + 1
01000 => [2,4] => 4 = 3 + 1
01001 => [2,3,1] => 3 = 2 + 1
01010 => [2,2,2] => 2 = 1 + 1
01011 => [2,2,1,1] => 2 = 1 + 1
01100 => [2,1,3] => 3 = 2 + 1
01101 => [2,1,2,1] => 2 = 1 + 1
01110 => [2,1,1,2] => 2 = 1 + 1
01111 => [2,1,1,1,1] => 2 = 1 + 1
10000 => [1,5] => 5 = 4 + 1
10001 => [1,4,1] => 4 = 3 + 1
10010 => [1,3,2] => 3 = 2 + 1
10011 => [1,3,1,1] => 3 = 2 + 1
Description
The largest part of an integer composition.
Matching statistic: St001235
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 1 = 0 + 1
1 => [1,1] => 2 = 1 + 1
00 => [3] => 1 = 0 + 1
01 => [2,1] => 2 = 1 + 1
10 => [1,2] => 2 = 1 + 1
11 => [1,1,1] => 3 = 2 + 1
000 => [4] => 1 = 0 + 1
001 => [3,1] => 2 = 1 + 1
010 => [2,2] => 2 = 1 + 1
011 => [2,1,1] => 3 = 2 + 1
100 => [1,3] => 2 = 1 + 1
101 => [1,2,1] => 2 = 1 + 1
110 => [1,1,2] => 3 = 2 + 1
111 => [1,1,1,1] => 4 = 3 + 1
0000 => [5] => 1 = 0 + 1
0001 => [4,1] => 2 = 1 + 1
0010 => [3,2] => 2 = 1 + 1
0011 => [3,1,1] => 3 = 2 + 1
0100 => [2,3] => 2 = 1 + 1
0101 => [2,2,1] => 2 = 1 + 1
0110 => [2,1,2] => 3 = 2 + 1
0111 => [2,1,1,1] => 4 = 3 + 1
1000 => [1,4] => 2 = 1 + 1
1001 => [1,3,1] => 2 = 1 + 1
1010 => [1,2,2] => 2 = 1 + 1
1011 => [1,2,1,1] => 3 = 2 + 1
1100 => [1,1,3] => 3 = 2 + 1
1101 => [1,1,2,1] => 3 = 2 + 1
1110 => [1,1,1,2] => 4 = 3 + 1
1111 => [1,1,1,1,1] => 5 = 4 + 1
00000 => [6] => 1 = 0 + 1
00001 => [5,1] => 2 = 1 + 1
00010 => [4,2] => 2 = 1 + 1
00011 => [4,1,1] => 3 = 2 + 1
00100 => [3,3] => 2 = 1 + 1
00101 => [3,2,1] => 2 = 1 + 1
00110 => [3,1,2] => 3 = 2 + 1
00111 => [3,1,1,1] => 4 = 3 + 1
01000 => [2,4] => 2 = 1 + 1
01001 => [2,3,1] => 2 = 1 + 1
01010 => [2,2,2] => 2 = 1 + 1
01011 => [2,2,1,1] => 3 = 2 + 1
01100 => [2,1,3] => 3 = 2 + 1
01101 => [2,1,2,1] => 3 = 2 + 1
01110 => [2,1,1,2] => 4 = 3 + 1
01111 => [2,1,1,1,1] => 5 = 4 + 1
10000 => [1,5] => 2 = 1 + 1
10001 => [1,4,1] => 2 = 1 + 1
10010 => [1,3,2] => 2 = 1 + 1
10011 => [1,3,1,1] => 3 = 2 + 1
Description
The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St000442
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => 2
01 => [2,1] => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 4
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 3
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001294
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 0
1 => [1,1] => [1,0,1,0]
 => 1
00 => [3] => [1,1,1,0,0,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 4
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2
Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]]. The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.
Matching statistic: St001296
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001296: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 0
1 => [1,1] => [1,0,1,0]
 => 1
00 => [3] => [1,1,1,0,0,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 4
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2
Description
The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].
Matching statistic: St000013
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ 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 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4 = 3 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4 = 3 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000147
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 2 = 1 + 1
1 => [1,1] => [1,1]
 => 1 = 0 + 1
00 => [3] => [3]
 => 3 = 2 + 1
01 => [2,1] => [2,1]
 => 2 = 1 + 1
10 => [1,2] => [2,1]
 => 2 = 1 + 1
11 => [1,1,1] => [1,1,1]
 => 1 = 0 + 1
000 => [4] => [4]
 => 4 = 3 + 1
001 => [3,1] => [3,1]
 => 3 = 2 + 1
010 => [2,2] => [2,2]
 => 2 = 1 + 1
011 => [2,1,1] => [2,1,1]
 => 2 = 1 + 1
100 => [1,3] => [3,1]
 => 3 = 2 + 1
101 => [1,2,1] => [2,1,1]
 => 2 = 1 + 1
110 => [1,1,2] => [2,1,1]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,1,1,1]
 => 1 = 0 + 1
0000 => [5] => [5]
 => 5 = 4 + 1
0001 => [4,1] => [4,1]
 => 4 = 3 + 1
0010 => [3,2] => [3,2]
 => 3 = 2 + 1
0011 => [3,1,1] => [3,1,1]
 => 3 = 2 + 1
0100 => [2,3] => [3,2]
 => 3 = 2 + 1
0101 => [2,2,1] => [2,2,1]
 => 2 = 1 + 1
0110 => [2,1,2] => [2,2,1]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [2,1,1,1]
 => 2 = 1 + 1
1000 => [1,4] => [4,1]
 => 4 = 3 + 1
1001 => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
1010 => [1,2,2] => [2,2,1]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [2,1,1,1]
 => 2 = 1 + 1
1100 => [1,1,3] => [3,1,1]
 => 3 = 2 + 1
1101 => [1,1,2,1] => [2,1,1,1]
 => 2 = 1 + 1
1110 => [1,1,1,2] => [2,1,1,1]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1 = 0 + 1
00000 => [6] => [6]
 => 6 = 5 + 1
00001 => [5,1] => [5,1]
 => 5 = 4 + 1
00010 => [4,2] => [4,2]
 => 4 = 3 + 1
00011 => [4,1,1] => [4,1,1]
 => 4 = 3 + 1
00100 => [3,3] => [3,3]
 => 3 = 2 + 1
00101 => [3,2,1] => [3,2,1]
 => 3 = 2 + 1
00110 => [3,1,2] => [3,2,1]
 => 3 = 2 + 1
00111 => [3,1,1,1] => [3,1,1,1]
 => 3 = 2 + 1
01000 => [2,4] => [4,2]
 => 4 = 3 + 1
01001 => [2,3,1] => [3,2,1]
 => 3 = 2 + 1
01010 => [2,2,2] => [2,2,2]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [2,2,1,1]
 => 2 = 1 + 1
01100 => [2,1,3] => [3,2,1]
 => 3 = 2 + 1
01101 => [2,1,2,1] => [2,2,1,1]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [2,2,1,1]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 2 = 1 + 1
10000 => [1,5] => [5,1]
 => 5 = 4 + 1
10001 => [1,4,1] => [4,1,1]
 => 4 = 3 + 1
10010 => [1,3,2] => [3,2,1]
 => 3 = 2 + 1
10011 => [1,3,1,1] => [3,1,1,1]
 => 3 = 2 + 1
Description
The largest part of an integer partition.
Matching statistic: St000444
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ 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 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4 = 3 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4 = 3 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000684
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
00 => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3 = 2 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
Description
The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$. The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$. One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$. Examples: * For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000686The finitistic dominant dimension of a Dyck path. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001530The depth of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000306The bounce count of a Dyck path. St000503The maximal difference between two elements in a common block. St000662The staircase size of the code of a permutation. St000956The maximal displacement of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001090The number of pop-stack-sorts needed to sort a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St000485The length of the longest cycle of a permutation. St000628The balance of a binary word. St000676The number of odd rises of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000734The last entry in the first row of a standard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001652The length of a longest interval of consecutive numbers. St001809The index of the step at the first peak of maximal height in a Dyck path. St000094The depth of an ordered tree. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000993The multiplicity of the largest part of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000772The multiplicity of the largest distance Laplacian eigenvalue in 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. St001589The nesting number of a perfect matching. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001568The smallest positive integer that does not appear twice in the partition. St000455The second largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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.