- FindStat

Your data matches 388 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000010
(load all 8 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ 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]
 => 2
00 => [3] => [3]
 => 1
01 => [2,1] => [2,1]
 => 2
10 => [1,2] => [2,1]
 => 2
11 => [1,1,1] => [1,1,1]
 => 3
000 => [4] => [4]
 => 1
001 => [3,1] => [3,1]
 => 2
010 => [2,2] => [2,2]
 => 2
100 => [1,3] => [3,1]
 => 2
111 => [1,1,1,1] => [1,1,1,1]
 => 4
0000 => [5] => [5]
 => 1
0001 => [4,1] => [4,1]
 => 2
0010 => [3,2] => [3,2]
 => 2
0100 => [2,3] => [3,2]
 => 2
1000 => [1,4] => [4,1]
 => 2
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5
00000 => [6] => [6]
 => 1
00001 => [5,1] => [5,1]
 => 2
00010 => [4,2] => [4,2]
 => 2
00100 => [3,3] => [3,3]
 => 2
01000 => [2,4] => [4,2]
 => 2
10000 => [1,5] => [5,1]
 => 2
11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 6
000001 => [6,1] => [6,1]
 => 2
000010 => [5,2] => [5,2]
 => 2
000100 => [4,3] => [4,3]
 => 2
001000 => [3,4] => [4,3]
 => 2
010000 => [2,5] => [5,2]
 => 2
100000 => [1,6] => [6,1]
 => 2
111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => 7
0000001 => [7,1] => [7,1]
 => 2
0000010 => [6,2] => [6,2]
 => 2
0000100 => [5,3] => [5,3]
 => 2
0001000 => [4,4] => [4,4]
 => 2
0010000 => [3,5] => [5,3]
 => 2
0100000 => [2,6] => [6,2]
 => 2
1000000 => [1,7] => [7,1]
 => 2
1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
 => 8
 => [1] => [1]
 => 1

Description

The length of the partition.

Matching statistic: St000011
(load all 6 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: 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]
 => 2
00 => [3] => [1,1,1,0,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => 2
10 => [1,2] => [1,0,1,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 2
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 2
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 2
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 2
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 2
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => 2
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => 2
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => 2
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => 2
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => 2
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 2
1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 2
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8
 => [1] => [1,0]
 => 1

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000097
(load all 13 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [2] => ([],2)
 => 1
1 => [1,1] => ([(0,1)],2)
 => 2
00 => [3] => ([],3)
 => 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
10 => [1,2] => ([(1,2)],3)
 => 2
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
000 => [4] => ([],4)
 => 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
100 => [1,3] => ([(2,3)],4)
 => 2
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
0000 => [5] => ([],5)
 => 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
1000 => [1,4] => ([(3,4)],5)
 => 2
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)
 => 5
00000 => [6] => ([],6)
 => 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2
01000 => [2,4] => ([(3,5),(4,5)],6)
 => 2
10000 => [1,5] => ([(4,5)],6)
 => 2
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)
 => 6
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2
001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => 2
010000 => [2,5] => ([(4,6),(5,6)],7)
 => 2
100000 => [1,6] => ([(5,6)],7)
 => 2
111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7
0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0000010 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0000100 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0001000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => 2
0100000 => [2,6] => ([(5,7),(6,7)],8)
 => 2
1000000 => [1,7] => ([(6,7)],8)
 => 2
1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => 8
 => [1] => ([],1)
 => 1

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000288
(load all 47 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [2] => 10 => 1
1 => [1,1] => 11 => 2
00 => [3] => 100 => 1
01 => [2,1] => 101 => 2
10 => [1,2] => 110 => 2
11 => [1,1,1] => 111 => 3
000 => [4] => 1000 => 1
001 => [3,1] => 1001 => 2
010 => [2,2] => 1010 => 2
100 => [1,3] => 1100 => 2
111 => [1,1,1,1] => 1111 => 4
0000 => [5] => 10000 => 1
0001 => [4,1] => 10001 => 2
0010 => [3,2] => 10010 => 2
0100 => [2,3] => 10100 => 2
1000 => [1,4] => 11000 => 2
1111 => [1,1,1,1,1] => 11111 => 5
00000 => [6] => 100000 => 1
00001 => [5,1] => 100001 => 2
00010 => [4,2] => 100010 => 2
00100 => [3,3] => 100100 => 2
01000 => [2,4] => 101000 => 2
10000 => [1,5] => 110000 => 2
11111 => [1,1,1,1,1,1] => 111111 => 6
000001 => [6,1] => 1000001 => 2
000010 => [5,2] => 1000010 => 2
000100 => [4,3] => 1000100 => 2
001000 => [3,4] => 1001000 => 2
010000 => [2,5] => 1010000 => 2
100000 => [1,6] => 1100000 => 2
111111 => [1,1,1,1,1,1,1] => 1111111 => 7
0000001 => [7,1] => 10000001 => 2
0000010 => [6,2] => 10000010 => 2
0000100 => [5,3] => 10000100 => 2
0001000 => [4,4] => 10001000 => 2
0010000 => [3,5] => 10010000 => 2
0100000 => [2,6] => 10100000 => 2
1000000 => [1,7] => 11000000 => 2
1111111 => [1,1,1,1,1,1,1,1] => 11111111 => 8
 => [1] => 1 => 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000381
(load all 15 compositions to match this statistic)

Mapping

Mp00105: Binary words —complement⟶ Binary words
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 => 1 => [1,1] => 1
1 => 0 => [2] => 2
00 => 11 => [1,1,1] => 1
01 => 10 => [1,2] => 2
10 => 01 => [2,1] => 2
11 => 00 => [3] => 3
000 => 111 => [1,1,1,1] => 1
001 => 110 => [1,1,2] => 2
010 => 101 => [1,2,1] => 2
100 => 011 => [2,1,1] => 2
111 => 000 => [4] => 4
0000 => 1111 => [1,1,1,1,1] => 1
0001 => 1110 => [1,1,1,2] => 2
0010 => 1101 => [1,1,2,1] => 2
0100 => 1011 => [1,2,1,1] => 2
1000 => 0111 => [2,1,1,1] => 2
1111 => 0000 => [5] => 5
00000 => 11111 => [1,1,1,1,1,1] => 1
00001 => 11110 => [1,1,1,1,2] => 2
00010 => 11101 => [1,1,1,2,1] => 2
00100 => 11011 => [1,1,2,1,1] => 2
01000 => 10111 => [1,2,1,1,1] => 2
10000 => 01111 => [2,1,1,1,1] => 2
11111 => 00000 => [6] => 6
000001 => 111110 => [1,1,1,1,1,2] => 2
000010 => 111101 => [1,1,1,1,2,1] => 2
000100 => 111011 => [1,1,1,2,1,1] => 2
001000 => 110111 => [1,1,2,1,1,1] => 2
010000 => 101111 => [1,2,1,1,1,1] => 2
100000 => 011111 => [2,1,1,1,1,1] => 2
111111 => 000000 => [7] => 7
0000001 => 1111110 => [1,1,1,1,1,1,2] => 2
0000010 => 1111101 => [1,1,1,1,1,2,1] => 2
0000100 => 1111011 => [1,1,1,1,2,1,1] => 2
0001000 => 1110111 => [1,1,1,2,1,1,1] => 2
0010000 => 1101111 => [1,1,2,1,1,1,1] => 2
0100000 => 1011111 => [1,2,1,1,1,1,1] => 2
1000000 => 0111111 => [2,1,1,1,1,1,1] => 2
1111111 => 0000000 => [8] => 8
 =>  => [1] => 1

Description

The largest part of an integer composition.

Matching statistic: St000808
(load all 15 compositions to match this statistic)

Mapping

Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => 1 => [1,1] => 1
1 => 0 => [2] => 2
00 => 11 => [1,1,1] => 1
01 => 10 => [1,2] => 2
10 => 01 => [2,1] => 2
11 => 00 => [3] => 3
000 => 111 => [1,1,1,1] => 1
001 => 110 => [1,1,2] => 2
010 => 101 => [1,2,1] => 2
100 => 011 => [2,1,1] => 2
111 => 000 => [4] => 4
0000 => 1111 => [1,1,1,1,1] => 1
0001 => 1110 => [1,1,1,2] => 2
0010 => 1101 => [1,1,2,1] => 2
0100 => 1011 => [1,2,1,1] => 2
1000 => 0111 => [2,1,1,1] => 2
1111 => 0000 => [5] => 5
00000 => 11111 => [1,1,1,1,1,1] => 1
00001 => 11110 => [1,1,1,1,2] => 2
00010 => 11101 => [1,1,1,2,1] => 2
00100 => 11011 => [1,1,2,1,1] => 2
01000 => 10111 => [1,2,1,1,1] => 2
10000 => 01111 => [2,1,1,1,1] => 2
11111 => 00000 => [6] => 6
000001 => 111110 => [1,1,1,1,1,2] => 2
000010 => 111101 => [1,1,1,1,2,1] => 2
000100 => 111011 => [1,1,1,2,1,1] => 2
001000 => 110111 => [1,1,2,1,1,1] => 2
010000 => 101111 => [1,2,1,1,1,1] => 2
100000 => 011111 => [2,1,1,1,1,1] => 2
111111 => 000000 => [7] => 7
0000001 => 1111110 => [1,1,1,1,1,1,2] => 2
0000010 => 1111101 => [1,1,1,1,1,2,1] => 2
0000100 => 1111011 => [1,1,1,1,2,1,1] => 2
0001000 => 1110111 => [1,1,1,2,1,1,1] => 2
0010000 => 1101111 => [1,1,2,1,1,1,1] => 2
0100000 => 1011111 => [1,2,1,1,1,1,1] => 2
1000000 => 0111111 => [2,1,1,1,1,1,1] => 2
1111111 => 0000000 => [8] => 8
 =>  => [1] => 1

Description

The number of up steps of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.

Matching statistic: St001777
(load all 3 compositions to match this statistic)

Mapping

Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => 0 => [2] => 0 = 1 - 1
1 => 1 => [1,1] => 1 = 2 - 1
00 => 00 => [3] => 0 = 1 - 1
01 => 01 => [2,1] => 1 = 2 - 1
10 => 01 => [2,1] => 1 = 2 - 1
11 => 11 => [1,1,1] => 2 = 3 - 1
000 => 000 => [4] => 0 = 1 - 1
001 => 001 => [3,1] => 1 = 2 - 1
010 => 001 => [3,1] => 1 = 2 - 1
100 => 001 => [3,1] => 1 = 2 - 1
111 => 111 => [1,1,1,1] => 3 = 4 - 1
0000 => 0000 => [5] => 0 = 1 - 1
0001 => 0001 => [4,1] => 1 = 2 - 1
0010 => 0001 => [4,1] => 1 = 2 - 1
0100 => 0001 => [4,1] => 1 = 2 - 1
1000 => 0001 => [4,1] => 1 = 2 - 1
1111 => 1111 => [1,1,1,1,1] => 4 = 5 - 1
00000 => 00000 => [6] => 0 = 1 - 1
00001 => 00001 => [5,1] => 1 = 2 - 1
00010 => 00001 => [5,1] => 1 = 2 - 1
00100 => 00001 => [5,1] => 1 = 2 - 1
01000 => 00001 => [5,1] => 1 = 2 - 1
10000 => 00001 => [5,1] => 1 = 2 - 1
11111 => 11111 => [1,1,1,1,1,1] => 5 = 6 - 1
000001 => 000001 => [6,1] => 1 = 2 - 1
000010 => 000001 => [6,1] => 1 = 2 - 1
000100 => 000001 => [6,1] => 1 = 2 - 1
001000 => 000001 => [6,1] => 1 = 2 - 1
010000 => 000001 => [6,1] => 1 = 2 - 1
100000 => 000001 => [6,1] => 1 = 2 - 1
111111 => 111111 => [1,1,1,1,1,1,1] => 6 = 7 - 1
0000001 => 0000001 => [7,1] => 1 = 2 - 1
0000010 => 0000001 => [7,1] => 1 = 2 - 1
0000100 => 0000001 => [7,1] => 1 = 2 - 1
0001000 => 0000001 => [7,1] => 1 = 2 - 1
0010000 => 0000001 => [7,1] => 1 = 2 - 1
0100000 => 0000001 => [7,1] => 1 = 2 - 1
1000000 => 0000001 => [7,1] => 1 = 2 - 1
1111111 => 1111111 => [1,1,1,1,1,1,1,1] => 7 = 8 - 1
 =>  => [1] => 0 = 1 - 1

Description

The number of weak descents in an integer composition. A weak descent of an integer composition

$\alpha=(a_1, \dots, a_n)$ is an index

$1\leq i < n$ such that

$a_i \geq a_{i+1}$ .

Matching statistic: St000013
(load all 5 compositions to match this statistic)

Mapping

Mp00105: Binary words —complement⟶ Binary words
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 => 1 => [1,1] => [1,0,1,0]
 => 1
1 => 0 => [2] => [1,1,0,0]
 => 2
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
11 => 00 => [3] => [1,1,1,0,0,0]
 => 3
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
11111 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
000001 => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
000010 => 111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
000100 => 111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
001000 => 110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
010000 => 101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
100000 => 011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => 2
111111 => 000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7
0000001 => 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
0000010 => 1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
0000100 => 1111011 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
0001000 => 1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
0010000 => 1101111 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
0100000 => 1011111 => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => 2
1000000 => 0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 2
1111111 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 8
 =>  => [1] => [1,0]
 => 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: St000098
(load all 13 compositions to match this statistic)

Mapping

Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => 0 => [2] => ([],2)
 => 1
1 => 1 => [1,1] => ([(0,1)],2)
 => 2
00 => 00 => [3] => ([],3)
 => 1
01 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
10 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
000 => 000 => [4] => ([],4)
 => 1
001 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
010 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
100 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
0000 => 0000 => [5] => ([],5)
 => 1
0001 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
0010 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
0100 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
1000 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
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)
 => 5
00000 => 00000 => [6] => ([],6)
 => 1
00001 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
00010 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
00100 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
01000 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
10000 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
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)
 => 6
000001 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
000010 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
000100 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
001000 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
010000 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
100000 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
111111 => 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7
0000001 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0000010 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0000100 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0001000 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0010000 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
0100000 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
1000000 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2
1111111 => 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => 8
 =>  => [1] => ([],1)
 => 1

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000147
(load all 2 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ 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] => [1,1]
 => [2]
 => 2
00 => [3] => [3]
 => [1,1,1]
 => 1
01 => [2,1] => [2,1]
 => [2,1]
 => 2
10 => [1,2] => [2,1]
 => [2,1]
 => 2
11 => [1,1,1] => [1,1,1]
 => [3]
 => 3
000 => [4] => [4]
 => [1,1,1,1]
 => 1
001 => [3,1] => [3,1]
 => [2,1,1]
 => 2
010 => [2,2] => [2,2]
 => [2,2]
 => 2
100 => [1,3] => [3,1]
 => [2,1,1]
 => 2
111 => [1,1,1,1] => [1,1,1,1]
 => [4]
 => 4
0000 => [5] => [5]
 => [1,1,1,1,1]
 => 1
0001 => [4,1] => [4,1]
 => [2,1,1,1]
 => 2
0010 => [3,2] => [3,2]
 => [2,2,1]
 => 2
0100 => [2,3] => [3,2]
 => [2,2,1]
 => 2
1000 => [1,4] => [4,1]
 => [2,1,1,1]
 => 2
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 5
00000 => [6] => [6]
 => [1,1,1,1,1,1]
 => 1
00001 => [5,1] => [5,1]
 => [2,1,1,1,1]
 => 2
00010 => [4,2] => [4,2]
 => [2,2,1,1]
 => 2
00100 => [3,3] => [3,3]
 => [2,2,2]
 => 2
01000 => [2,4] => [4,2]
 => [2,2,1,1]
 => 2
10000 => [1,5] => [5,1]
 => [2,1,1,1,1]
 => 2
11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [6]
 => 6
000001 => [6,1] => [6,1]
 => [2,1,1,1,1,1]
 => 2
000010 => [5,2] => [5,2]
 => [2,2,1,1,1]
 => 2
000100 => [4,3] => [4,3]
 => [2,2,2,1]
 => 2
001000 => [3,4] => [4,3]
 => [2,2,2,1]
 => 2
010000 => [2,5] => [5,2]
 => [2,2,1,1,1]
 => 2
100000 => [1,6] => [6,1]
 => [2,1,1,1,1,1]
 => 2
111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => [7]
 => 7
0000001 => [7,1] => [7,1]
 => [2,1,1,1,1,1,1]
 => 2
0000010 => [6,2] => [6,2]
 => [2,2,1,1,1,1]
 => 2
0000100 => [5,3] => [5,3]
 => [2,2,2,1,1]
 => 2
0001000 => [4,4] => [4,4]
 => [2,2,2,2]
 => 2
0010000 => [3,5] => [5,3]
 => [2,2,2,1,1]
 => 2
0100000 => [2,6] => [6,2]
 => [2,2,1,1,1,1]
 => 2
1000000 => [1,7] => [7,1]
 => [2,1,1,1,1,1,1]
 => 2
1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
 => [8]
 => 8
 => [1] => [1]
 => [1]
 => 1

Description

The largest part of an integer partition.

The following 378 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000479The Ramsey number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000733The row containing the largest entry of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000784The maximum of the length and the largest part of the integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000982The length of the longest constant subword. St001267The length of the Lyndon factorization of the binary word. St001330The hat guessing number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001884The number of borders of a binary word. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000519The largest length of a factor maximising the subword complexity. St000532The total number of rook placements on a Ferrers board. St000548The number of different non-empty partial sums of an integer partition. St001176The size of a partition minus its first part. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001400The total number of Littlewood-Richardson tableaux of given shape. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St001581The achromatic number of a graph. St000877The depth of the binary word interpreted as a path. St000258The burning number of a graph. St000395The sum of the heights of the peaks of a Dyck path. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001523The degree of symmetry of a Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001672The restrained domination number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000012The area of a Dyck path. St000391The sum of the positions of the ones in a binary word. St000439The position of the first down step of a Dyck path. St000778The metric dimension of a graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001658The total number of rook placements on a Ferrers board. St001721The degree of a binary word. St001949The rigidity index of a graph. St000007The number of saliances of the permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000144The pyramid weight of the Dyck path. St000153The number of adjacent cycles of a permutation. St000444The length of the maximal rise of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000546The number of global descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000984The number of boxes below precisely one peak. St000996The number of exclusive left-to-right maxima of a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000237The number of small exceedances. St000167The number of leaves of an ordered tree. St000925The number of topologically connected components of a set partition. St001058The breadth of the ordered tree. St000806The semiperimeter of the associated bargraph. St001809The index of the step at the first peak of maximal height in a Dyck path. St000159The number of distinct parts of the integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St001933The largest multiplicity of a part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000874The position of the last double rise in a Dyck path. St001644The dimension of a graph. St000931The number of occurrences of the pattern UUU in a Dyck path. St000172The Grundy number of a graph. St000686The finitistic dominant dimension of a Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. 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: St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000025The number of initial rises of a Dyck path. St000058The order of a permutation. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000363The number of minimal vertex covers of a graph. St000383The last part of an integer composition. St000678The number of up steps after the last double rise of a Dyck path. St000722The number of different neighbourhoods in a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001883The mutual visibility number of a graph. St000024The number of double up and double down steps of a Dyck path. St000171The degree of the graph. St000234The number of global ascents of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000454The largest eigenvalue of a graph if it is integral. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000164The number of short pairs. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000390The number of runs of ones in a binary word. St000822The Hadwiger number of the graph. St000843The decomposition number of a perfect matching. St001316The domatic number of a graph. St001725The harmonious chromatic number of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000292The number of ascents of a binary word. St001270The bandwidth of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001962The proper pathwidth of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001645The pebbling number of a connected graph. St000209Maximum difference of elements in cycles. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St001812The biclique partition number of a graph. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000809The reduced reflection length of the permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000456The monochromatic index of a connected graph. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001175The size of a partition minus the hook length of the base cell. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000141The maximum drop size of a permutation. St000308The height of the tree associated to a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000451The length of the longest pattern of the form k 1 2. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000015The number of peaks of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001202Call 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. St001530The depth of a Dyck path. 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. 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

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000155The number of exceedances (also excedences) of a permutation. St000224The sorting index of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000264The girth of a graph, which is not a tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000741The Colin de Verdière graph invariant. St000864The number of circled entries of the shifted recording tableau of a permutation. St001119The length of a shortest maximal path in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001702The absolute value of the determinant of the adjacency matrix of a graph. St000225Difference between largest and smallest parts in a partition. St000061The number of nodes on the left branch of a binary tree. St000464The Schultz index of a connected graph. 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

$eA$ . 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

$eA$ . 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

$eA$ . St001545The second Elser number of a connected graph. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001498The normalised height of a Nakayama algebra with magnitude 1. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000455The second largest eigenvalue of a graph if it is integral. St000993The multiplicity of the largest part of an integer partition. St000316The number of non-left-to-right-maxima of a permutation. St000702The number of weak deficiencies of a permutation. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001684The reduced word complexity of a permutation. St001727The number of invisible inversions of a permutation. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001060The distinguishing index of a graph. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000699The toughness times the least common multiple of 1,. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St000284The Plancherel distribution on integer partitions. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001568The smallest positive integer that does not appear twice in the partition. St001712The number of natural descents of a standard Young tableau. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001570The minimal number of edges to add to make a graph Hamiltonian. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2.