Your data matches 469 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001392
(load all 14 compositions to match this statistic)
Mapping
St001392: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 2
[2,1]
 => 0
[1,1,1]
 => 0
[4]
 => 3
[3,1]
 => 2
[2,2]
 => 1
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 4
[4,1]
 => 3
[3,2]
 => 1
[3,1,1]
 => 2
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000462
(load all 10 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00129: Dyck paths ā€”to 321-avoiding permutation (Billey-Jockusch-Stanley)āŸ¶ Permutations
St000462: Permutations 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]
 => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 2
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 3
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => 4
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 0
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000008
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00129: Dyck paths ā€”to 321-avoiding permutation (Billey-Jockusch-Stanley)āŸ¶ Permutations
Mp00248: Permutations ā€”DEX compositionāŸ¶ Integer compositions
St000008: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2] => 0
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => [1,2] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => [3] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,2] => 2
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => [3] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [4] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [3,2] => 3
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4] => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [5] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => [4,2] => 4
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [3,2] => 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,3] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [4] => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5] => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [6] => 0
Description
The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000034
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00025: Dyck paths ā€”to 132-avoiding permutationāŸ¶ Permutations
Mp00086: Permutations ā€”first fundamental transformationāŸ¶ Permutations
St000034: Permutations 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]
 => [2,1] => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3,1,2] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1,4,2] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,4,3,1] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,4,5,2] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [2,4,1,3] => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,3,1,2] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,1,3,2] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,3,2,4,1] => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 4
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St000217
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00024: Dyck paths ā€”to 321-avoiding permutationāŸ¶ Permutations
Mp00149: Permutations ā€”Lehmer code rotationāŸ¶ Permutations
St000217: Permutations 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]
 => [2,1] => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,3,4,2] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => [3,2,1] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [3,4,1,2] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,3,4,5,2] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [4,2,3,1] => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,4,2,1] => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,3,4,5,6,2] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [5,2,3,4,1] => 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [4,2,1,3] => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [3,2,4,1] => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [3,4,5,2,1] => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [3,4,5,6,1,2] => 4
Description
The number of occurrences of the pattern 312 in a permutation.
Matching statistic: St000218
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00129: Dyck paths ā€”to 321-avoiding permutation (Billey-Jockusch-Stanley)āŸ¶ Permutations
Mp00175: Permutations ā€”inverse Foata bijectionāŸ¶ Permutations
St000218: Permutations 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]
 => [2,1] => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => [3,1,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [4,1,2,3] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,3,1] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [1,3,4,2] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [3,1,2,4] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [1,2,4,5,3] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [3,4,1,2] => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,4,1,3] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,3,1,4] => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,5,3,4] => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 4
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000290
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00031: Dyck paths ā€”to 312-avoiding permutationāŸ¶ Permutations
Mp00109: Permutations ā€”descent wordāŸ¶ Binary words
St000290: Binary words 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]
 => [1,2] => 0 => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 01 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 110 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 011 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1110 => 3
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0111 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 11110 => 4
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 0110 => 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 0011 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 01111 => 0
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000293
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00023: Dyck paths ā€”to non-crossing permutationāŸ¶ Permutations
Mp00114: Permutations ā€”connectivity setāŸ¶ Binary words
St000293: Binary words 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]
 => [1,2] => 1 => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 01 => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 10 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 001 => 0
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 11 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 100 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0001 => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 001 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 010 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 100 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1000 => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 00001 => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => 0001 => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 011 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 101 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 110 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1000 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 10000 => 4
Description
The number of inversions of a binary word.
Matching statistic: St000355
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00101: Dyck paths ā€”decomposition reverseāŸ¶ Dyck paths
Mp00023: Dyck paths ā€”to non-crossing permutationāŸ¶ Permutations
St000355: Permutations 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]
 => [1,1,0,0]
 => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 3
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 4
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => 0
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000369
(load all 7 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00327: Dyck paths ā€”inverse Kreweras complementāŸ¶ Dyck paths
Mp00124: Dyck paths ā€”Adin-Bagno-Roichman transformationāŸ¶ Dyck paths
St000369: 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]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4
Description
The dinv deficit of a Dyck path. For a Dyck path $D$ of semilength $n$, this is defined as $$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$ In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$. See also [[St000376]] for the bounce deficit.
The following 459 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000682The Grundy value of Welter's game on a binary word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001377The major index minus the number of inversions of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001841The number of inversions of a set partition. St000018The number of inversions of a permutation. St000026The position of the first return of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000472The sum of the ascent bottoms of a permutation. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000839The largest opener of a set partition. St000883The number of longest increasing subsequences of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000988The orbit size of a permutation under Foata's bijection. St001090The number of pop-stack-sorts needed to sort a permutation. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001463The number of distinct columns in the nullspace of a graph. St000219The number of occurrences of the pattern 231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000877The depth of the binary word interpreted as a path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001438The number of missing boxes of a skew partition. St001557The number of inversions of the second entry of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000060The greater neighbor of the maximum. St000133The "bounce" of a permutation. 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. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000054The first entry of the permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001279The sum of the parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000169The cocharge of a standard tableau. St000305The inverse major index of a permutation. St000330The (standard) major index of a standard tableau. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000653The last descent of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000798The makl of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000961The shifted major index of a permutation. St000979Half of MacMahon's equal index of a Dyck path. St001161The major index north count of a Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001480The number of simple summands of the module J^2/J^3. St001541The Gini index of an integer partition. St001657The number of twos in an integer partition. St001671Haglund's hag of a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001759The Rajchgot index of a permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001956The comajor index for set-valued two-row standard Young tableaux. St000326The position of the first one in a binary word after appending a 1 at the end. St000454The largest eigenvalue of a graph if it is integral. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001556The number of inversions of the third entry of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000654The first descent of a permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000477The weight of a partition according to Alladi. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000383The last part of an integer composition. St000456The monochromatic index of a connected 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. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. 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. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. 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. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000295The length of the border of a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000461The rix statistic of a permutation. St000471The sum of the ascent tops of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000503The maximal difference between two elements in a common block. St000538The number of even inversions of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000572The dimension exponent of a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000646The number of big ascents of a permutation. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000711The number of big exceedences of a permutation. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000747A variant of the major index of a set partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000873The aix statistic of a permutation. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. 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. St001371The length of the longest Yamanouchi prefix of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001721The degree of a binary word. St001930The weak major index of a binary word. St001960The number of descents of a permutation minus one if its first entry is not one. St000422The energy of a graph, if it is integral. St001330The hat guessing number of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000012The area of a Dyck path. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000143The largest repeated part of a partition. St000145The Dyson rank of a partition. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000228The size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000306The bounce count of a Dyck path. 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. St000377The dinv defect of an integer partition. 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. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000658The number of rises of length 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. 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. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000944The 3-degree of an integer partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000992The alternating sum of the parts of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules 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. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in 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. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001214The aft of an integer partition. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001248Sum of the even parts of a partition. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001423The number of distinct cubes in a binary word. St001484The number of singletons of an integer partition. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. 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. St001524The degree of symmetry of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001955The number of natural descents for set-valued two row standard Young tableaux. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000259The diameter of a connected graph. St001175The size of a partition minus the hook length of the base cell. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St000352The Elizalde-Pak rank of a permutation. St001645The pebbling number of a connected graph. St000007The number of saliances of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000091The descent variation of a composition. St000365The number of double ascents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000709The number of occurrences of 14-2-3 or 14-3-2. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001435The number of missing boxes in the first row. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000090The variation of a composition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St001151The number of blocks with odd minimum. St001487The number of inner corners of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St001375The pancake length of a permutation. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000137The Grundy value of an integer partition. St000177The number of free tiles in the pattern. St000178Number of free entries. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000509The diagonal index (content) of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000678The number of up steps after the last double rise of a Dyck path. 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. St000744The length of the path to the largest entry in a standard Young tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000981The length of the longest zigzag subpath. St000997The even-odd crank of an integer partition. St001060The distinguishing index of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001118The acyclic chromatic index of a graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001204Call 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001520The number of strict 3-descents. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001561The value of the elementary symmetric function evaluated at 1. 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. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001808The box weight or horizontal decoration of a Dyck path. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001948The number of augmented double ascents of a permutation. St001959The product of the heights of the peaks of a Dyck path. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000307The number of rowmotion orbits of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000717The number of ordinal summands of a poset.