Your data matches 316 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000382
(load all 10 compositions to match this statistic)
Mapping
St000382: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => 1 = 0 + 1
[1,1] => 1 = 0 + 1
[2] => 2 = 1 + 1
[1,1,1] => 1 = 0 + 1
[1,2] => 1 = 0 + 1
[2,1] => 2 = 1 + 1
[3] => 3 = 2 + 1
[1,1,1,1] => 1 = 0 + 1
[1,1,2] => 1 = 0 + 1
[1,2,1] => 1 = 0 + 1
[1,3] => 1 = 0 + 1
[2,1,1] => 2 = 1 + 1
[2,2] => 2 = 1 + 1
[3,1] => 3 = 2 + 1
[4] => 4 = 3 + 1
[1,1,1,1,1] => 1 = 0 + 1
[1,1,1,2] => 1 = 0 + 1
[1,1,2,1] => 1 = 0 + 1
[1,1,3] => 1 = 0 + 1
[1,2,1,1] => 1 = 0 + 1
[1,2,2] => 1 = 0 + 1
[1,3,1] => 1 = 0 + 1
[1,4] => 1 = 0 + 1
[2,1,1,1] => 2 = 1 + 1
[2,1,2] => 2 = 1 + 1
[2,2,1] => 2 = 1 + 1
[2,3] => 2 = 1 + 1
[3,1,1] => 3 = 2 + 1
[3,2] => 3 = 2 + 1
[4,1] => 4 = 3 + 1
[5] => 5 = 4 + 1
[1,1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,2] => 1 = 0 + 1
[1,1,1,2,1] => 1 = 0 + 1
[1,1,1,3] => 1 = 0 + 1
[1,1,2,1,1] => 1 = 0 + 1
[1,1,2,2] => 1 = 0 + 1
[1,1,3,1] => 1 = 0 + 1
[1,1,4] => 1 = 0 + 1
[1,2,1,1,1] => 1 = 0 + 1
[1,2,1,2] => 1 = 0 + 1
[1,2,2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,1,1] => 1 = 0 + 1
[1,3,2] => 1 = 0 + 1
[1,4,1] => 1 = 0 + 1
[1,5] => 1 = 0 + 1
[2,1,1,1,1] => 2 = 1 + 1
[2,1,1,2] => 2 = 1 + 1
[2,1,2,1] => 2 = 1 + 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 14 compositions to match this statistic)
Mapping
St000383: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => 1 = 0 + 1
[1,1] => 1 = 0 + 1
[2] => 2 = 1 + 1
[1,1,1] => 1 = 0 + 1
[1,2] => 2 = 1 + 1
[2,1] => 1 = 0 + 1
[3] => 3 = 2 + 1
[1,1,1,1] => 1 = 0 + 1
[1,1,2] => 2 = 1 + 1
[1,2,1] => 1 = 0 + 1
[1,3] => 3 = 2 + 1
[2,1,1] => 1 = 0 + 1
[2,2] => 2 = 1 + 1
[3,1] => 1 = 0 + 1
[4] => 4 = 3 + 1
[1,1,1,1,1] => 1 = 0 + 1
[1,1,1,2] => 2 = 1 + 1
[1,1,2,1] => 1 = 0 + 1
[1,1,3] => 3 = 2 + 1
[1,2,1,1] => 1 = 0 + 1
[1,2,2] => 2 = 1 + 1
[1,3,1] => 1 = 0 + 1
[1,4] => 4 = 3 + 1
[2,1,1,1] => 1 = 0 + 1
[2,1,2] => 2 = 1 + 1
[2,2,1] => 1 = 0 + 1
[2,3] => 3 = 2 + 1
[3,1,1] => 1 = 0 + 1
[3,2] => 2 = 1 + 1
[4,1] => 1 = 0 + 1
[5] => 5 = 4 + 1
[1,1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,2] => 2 = 1 + 1
[1,1,1,2,1] => 1 = 0 + 1
[1,1,1,3] => 3 = 2 + 1
[1,1,2,1,1] => 1 = 0 + 1
[1,1,2,2] => 2 = 1 + 1
[1,1,3,1] => 1 = 0 + 1
[1,1,4] => 4 = 3 + 1
[1,2,1,1,1] => 1 = 0 + 1
[1,2,1,2] => 2 = 1 + 1
[1,2,2,1] => 1 = 0 + 1
[1,2,3] => 3 = 2 + 1
[1,3,1,1] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[1,4,1] => 1 = 0 + 1
[1,5] => 5 = 4 + 1
[2,1,1,1,1] => 1 = 0 + 1
[2,1,1,2] => 2 = 1 + 1
[2,1,2,1] => 1 = 0 + 1
Description
The last part of an integer composition.
Matching statistic: St000025
(load all 16 compositions to match this statistic)
Mapping
Mp00231: Integer compositions ā€”bounce pathāŸ¶ Dyck paths
St000025: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 1 = 0 + 1
[1,1] => [1,0,1,0]
 => 1 = 0 + 1
[2] => [1,1,0,0]
 => 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
(load all 9 compositions to match this statistic)
Mapping
Mp00231: Integer compositions ā€”bounce pathāŸ¶ Dyck paths
St000026: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 1 = 0 + 1
[1,1] => [1,0,1,0]
 => 1 = 0 + 1
[2] => [1,1,0,0]
 => 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000297
(load all 11 compositions to match this statistic)
Mapping
Mp00094: Integer compositions ā€”to binary wordāŸ¶ Binary words
St000297: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1 = 0 + 1
[1,1] => 11 => 2 = 1 + 1
[2] => 10 => 1 = 0 + 1
[1,1,1] => 111 => 3 = 2 + 1
[1,2] => 110 => 2 = 1 + 1
[2,1] => 101 => 1 = 0 + 1
[3] => 100 => 1 = 0 + 1
[1,1,1,1] => 1111 => 4 = 3 + 1
[1,1,2] => 1110 => 3 = 2 + 1
[1,2,1] => 1101 => 2 = 1 + 1
[1,3] => 1100 => 2 = 1 + 1
[2,1,1] => 1011 => 1 = 0 + 1
[2,2] => 1010 => 1 = 0 + 1
[3,1] => 1001 => 1 = 0 + 1
[4] => 1000 => 1 = 0 + 1
[1,1,1,1,1] => 11111 => 5 = 4 + 1
[1,1,1,2] => 11110 => 4 = 3 + 1
[1,1,2,1] => 11101 => 3 = 2 + 1
[1,1,3] => 11100 => 3 = 2 + 1
[1,2,1,1] => 11011 => 2 = 1 + 1
[1,2,2] => 11010 => 2 = 1 + 1
[1,3,1] => 11001 => 2 = 1 + 1
[1,4] => 11000 => 2 = 1 + 1
[2,1,1,1] => 10111 => 1 = 0 + 1
[2,1,2] => 10110 => 1 = 0 + 1
[2,2,1] => 10101 => 1 = 0 + 1
[2,3] => 10100 => 1 = 0 + 1
[3,1,1] => 10011 => 1 = 0 + 1
[3,2] => 10010 => 1 = 0 + 1
[4,1] => 10001 => 1 = 0 + 1
[5] => 10000 => 1 = 0 + 1
[1,1,1,1,1,1] => 111111 => 6 = 5 + 1
[1,1,1,1,2] => 111110 => 5 = 4 + 1
[1,1,1,2,1] => 111101 => 4 = 3 + 1
[1,1,1,3] => 111100 => 4 = 3 + 1
[1,1,2,1,1] => 111011 => 3 = 2 + 1
[1,1,2,2] => 111010 => 3 = 2 + 1
[1,1,3,1] => 111001 => 3 = 2 + 1
[1,1,4] => 111000 => 3 = 2 + 1
[1,2,1,1,1] => 110111 => 2 = 1 + 1
[1,2,1,2] => 110110 => 2 = 1 + 1
[1,2,2,1] => 110101 => 2 = 1 + 1
[1,2,3] => 110100 => 2 = 1 + 1
[1,3,1,1] => 110011 => 2 = 1 + 1
[1,3,2] => 110010 => 2 = 1 + 1
[1,4,1] => 110001 => 2 = 1 + 1
[1,5] => 110000 => 2 = 1 + 1
[2,1,1,1,1] => 101111 => 1 = 0 + 1
[2,1,1,2] => 101110 => 1 = 0 + 1
[2,1,2,1] => 101101 => 1 = 0 + 1
Description
The number of leading ones in a binary word.
Matching statistic: St001135
(load all 5 compositions to match this statistic)
Mapping
Mp00231: Integer compositions ā€”bounce pathāŸ¶ Dyck paths
St001135: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 1 = 0 + 1
[1,1] => [1,0,1,0]
 => 2 = 1 + 1
[2] => [1,1,0,0]
 => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000439
(load all 16 compositions to match this statistic)
Mapping
Mp00231: Integer compositions ā€”bounce pathāŸ¶ Dyck paths
St000439: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 2 = 0 + 2
[1,1] => [1,0,1,0]
 => 2 = 0 + 2
[2] => [1,1,0,0]
 => 3 = 1 + 2
[1,1,1] => [1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[2,1] => [1,1,0,0,1,0]
 => 3 = 1 + 2
[3] => [1,1,1,0,0,0]
 => 4 = 2 + 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
[2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
[3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
[4] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 2 + 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 3 + 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 0 + 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 0 + 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 0 + 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 1 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St001176
(load all 6 compositions to match this statistic)
Mapping
Mp00184: Integer compositions ā€”to threshold graphāŸ¶ Graphs
Mp00037: Graphs ā€”to partition of connected componentsāŸ¶ Integer partitions
St001176: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 0
[1,1] => ([(0,1)],2)
 => [2]
 => 0
[2] => ([],2)
 => [1,1]
 => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
[1,2] => ([(1,2)],3)
 => [2,1]
 => 1
[2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 0
[3] => ([],3)
 => [1,1,1]
 => 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
[1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
[2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 0
[4] => ([],4)
 => [1,1,1,1]
 => 3
[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]
 => 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 3
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0
[5] => ([],5)
 => [1,1,1,1,1]
 => 4
[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]
 => 0
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 2
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 3
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
[1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 4
[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)
 => [6]
 => 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001657
(load all 7 compositions to match this statistic)
Mapping
Mp00180: Integer compositions ā€”to ribbonāŸ¶ Skew partitions
Mp00182: Skew partitions ā€”outer shapeāŸ¶ Integer partitions
St001657: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => [1]
 => 0
[1,1] => [[1,1],[]]
 => [1,1]
 => 0
[2] => [[2],[]]
 => [2]
 => 1
[1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => 0
[1,2] => [[2,1],[]]
 => [2,1]
 => 1
[2,1] => [[2,2],[1]]
 => [2,2]
 => 2
[3] => [[3],[]]
 => [3]
 => 0
[1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => 0
[1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => 2
[1,3] => [[3,1],[]]
 => [3,1]
 => 0
[2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 3
[2,2] => [[3,2],[1]]
 => [3,2]
 => 1
[3,1] => [[3,3],[2]]
 => [3,3]
 => 0
[4] => [[4],[]]
 => [4]
 => 0
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => 0
[1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => 2
[1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 3
[1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 0
[1,4] => [[4,1],[]]
 => [4,1]
 => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 4
[2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 1
[2,3] => [[4,2],[1]]
 => [4,2]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => 0
[3,2] => [[4,3],[2]]
 => [4,3]
 => 0
[4,1] => [[4,4],[3]]
 => [4,4]
 => 0
[5] => [[5],[]]
 => [5]
 => 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => [1,1,1,1,1,1]
 => 0
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => [2,1,1,1,1]
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [2,2,1,1,1]
 => 2
[1,1,1,3] => [[3,1,1,1],[]]
 => [3,1,1,1]
 => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [2,2,2,1,1]
 => 3
[1,1,2,2] => [[3,2,1,1],[1]]
 => [3,2,1,1]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [3,3,1,1]
 => 0
[1,1,4] => [[4,1,1],[]]
 => [4,1,1]
 => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [2,2,2,2,1]
 => 4
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [3,2,2,1]
 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [3,3,2,1]
 => 1
[1,2,3] => [[4,2,1],[1]]
 => [4,2,1]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [3,3,3,1]
 => 0
[1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => 0
[1,4,1] => [[4,4,1],[3]]
 => [4,4,1]
 => 0
[1,5] => [[5,1],[]]
 => [5,1]
 => 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [2,2,2,2,2]
 => 5
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [3,3,2,2]
 => 2
Description
The number of twos in an integer partition. The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000010
(load all 3 compositions to match this statistic)
Mapping
Mp00184: Integer compositions ā€”to threshold graphāŸ¶ Graphs
Mp00037: Graphs ā€”to partition of connected componentsāŸ¶ Integer partitions
St000010: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 1 = 0 + 1
[1,1] => ([(0,1)],2)
 => [2]
 => 1 = 0 + 1
[2] => ([],2)
 => [1,1]
 => 2 = 1 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
[1,2] => ([(1,2)],3)
 => [2,1]
 => 2 = 1 + 1
[2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
[3] => ([],3)
 => [1,1,1]
 => 3 = 2 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
[1,3] => ([(2,3)],4)
 => [2,1,1]
 => 3 = 2 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
[2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
[4] => ([],4)
 => [1,1,1,1]
 => 4 = 3 + 1
[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]
 => 1 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 2 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
[1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 4 = 3 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
[2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 2 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
[5] => ([],5)
 => [1,1,1,1,1]
 => 5 = 4 + 1
[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]
 => 1 = 0 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 3 = 2 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 4 = 3 + 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 3 = 2 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
[1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 5 = 4 + 1
[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)
 => [6]
 => 1 = 0 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 1 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
Description
The length of the partition.
The following 306 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000011The number of touch points (or returns) of a Dyck path. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000273The domination number of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000505The biggest entry in the block containing the 1. St000544The cop number of a graph. St000759The smallest missing part in an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000008The major index of the composition. St000012The area of a Dyck path. St000053The number of valleys of the Dyck path. St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000536The pathwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001777The number of weak descents in an integer composition. St001910The height of the middle non-run of a Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000532The total number of rook placements on a Ferrers board. St000700The protection number of an ordered tree. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001826The maximal number of leaves on a vertex of a graph. St000363The number of minimal vertex covers of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000287The number of connected components of a graph. St000504The cardinality of the first block of a set partition. St000553The number of blocks of a graph. St000823The number of unsplittable factors of the set partition. St000917The open packing number of a graph. St001672The restrained domination number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000932The number of occurrences of the pattern UDU in a Dyck path. St000984The number of boxes below precisely one peak. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000286The number of connected components of the complement of a graph. St000617The number of global maxima of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000729The minimal arc length of a set partition. St000822The Hadwiger number of the graph. St001038The minimal height of a column in the parallelogram polyomino associated with 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. St001316The domatic number of a graph. St001963The tree-depth of a graph. St001812The biclique partition number of a graph. St001827The number of two-component spanning forests of a graph. St001828The Euler characteristic of a graph. St001479The number of bridges of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000054The first entry of the permutation. St000993The multiplicity of the largest part of an integer partition. St001330The hat guessing number of a graph. St000295The length of the border of a binary word. St000877The depth of the binary word interpreted as a path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000546The number of global descents of a permutation. St000883The number of longest increasing subsequences of a permutation. St000989The number of final rises of a permutation. St000501The size of the first part in the decomposition of a permutation. St000237The number of small exceedances. St000454The largest eigenvalue of a graph if it is integral. St000542The number of left-to-right-minima of a permutation. St000740The last entry 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. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000214The number of adjacencies of a permutation. St000007The number of saliances of the permutation. St000674The number of hills of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000441The number of successions of a permutation. St000203The number of external nodes of a binary tree. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001587Half of the largest even part of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St000734The last entry in the first row of a standard tableau. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000699The toughness times the least common multiple of 1,. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000258The burning number of a graph. 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. St000133The "bounce" of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. 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. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001360The number of covering relations in Young's lattice below a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. 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. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001933The largest multiplicity of a part in an integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000352The Elizalde-Pak rank of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000061The number of nodes on the left branch of a binary tree. St001340The cardinality of a minimal non-edge isolating set of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001480The number of simple summands of the module J^2/J^3. St000918The 2-limited packing number of a graph. St001462The number of factors of a standard tableaux under concatenation. St000455The second largest eigenvalue of a graph if it is integral. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000731The number of double exceedences of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000137The Grundy value of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001961The sum of the greatest common divisors of all pairs of parts. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000338The number of pixed points of a permutation. St000456The monochromatic index of a connected graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000241The number of cyclical small excedances. St000284The Plancherel distribution on integer partitions. St000315The number of isolated vertices of a graph. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000060The greater neighbor of the maximum. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001530The depth of a Dyck path. St000756The sum of the positions of the left to right maxima of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001545The second Elser number of a connected graph. St001615The number of join prime elements of a lattice. St001712The number of natural descents of a standard Young tableau. St001651The Frankl number of a lattice. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001568The smallest positive integer that does not appear twice in the partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001557The number of inversions of the second entry of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.