Your data matches 351 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000290
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
Mp00094: Integer compositions ā€”to binary 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
0 => [2] => 10 => 1 = 0 + 1
1 => [1,1] => 11 => 0 = -1 + 1
00 => [3] => 100 => 1 = 0 + 1
01 => [2,1] => 101 => 1 = 0 + 1
11 => [1,1,1] => 111 => 0 = -1 + 1
000 => [4] => 1000 => 1 = 0 + 1
001 => [3,1] => 1001 => 1 = 0 + 1
011 => [2,1,1] => 1011 => 1 = 0 + 1
111 => [1,1,1,1] => 1111 => 0 = -1 + 1
0000 => [5] => 10000 => 1 = 0 + 1
0001 => [4,1] => 10001 => 1 = 0 + 1
0011 => [3,1,1] => 10011 => 1 = 0 + 1
0111 => [2,1,1,1] => 10111 => 1 = 0 + 1
1111 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
00000 => [6] => 100000 => 1 = 0 + 1
00001 => [5,1] => 100001 => 1 = 0 + 1
00011 => [4,1,1] => 100011 => 1 = 0 + 1
00111 => [3,1,1,1] => 100111 => 1 = 0 + 1
01111 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
11111 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
=> [1] => 1 => 0 = -1 + 1
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000291
(load all 8 compositions to match this statistic)
Mapping
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
Mp00094: Integer compositions ā€”to binary wordāŸ¶ Binary words
St000291: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1 = 0 + 1
1 => [1,1] => 11 => 0 = -1 + 1
00 => [3] => 100 => 1 = 0 + 1
01 => [2,1] => 101 => 1 = 0 + 1
11 => [1,1,1] => 111 => 0 = -1 + 1
000 => [4] => 1000 => 1 = 0 + 1
001 => [3,1] => 1001 => 1 = 0 + 1
011 => [2,1,1] => 1011 => 1 = 0 + 1
111 => [1,1,1,1] => 1111 => 0 = -1 + 1
0000 => [5] => 10000 => 1 = 0 + 1
0001 => [4,1] => 10001 => 1 = 0 + 1
0011 => [3,1,1] => 10011 => 1 = 0 + 1
0111 => [2,1,1,1] => 10111 => 1 = 0 + 1
1111 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
00000 => [6] => 100000 => 1 = 0 + 1
00001 => [5,1] => 100001 => 1 = 0 + 1
00011 => [4,1,1] => 100011 => 1 = 0 + 1
00111 => [3,1,1,1] => 100111 => 1 = 0 + 1
01111 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
11111 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
=> [1] => 1 => 0 = -1 + 1
Description
The number of descents of a binary word.
Matching statistic: St000480
(load all 5 compositions to match this statistic)
Mapping
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
Mp00040: Integer compositions ā€”to partitionāŸ¶ Integer partitions
St000480: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 1 = 0 + 1
1 => [1,1] => [1,1]
 => 0 = -1 + 1
00 => [3] => [3]
 => 1 = 0 + 1
01 => [2,1] => [2,1]
 => 1 = 0 + 1
11 => [1,1,1] => [1,1,1]
 => 0 = -1 + 1
000 => [4] => [4]
 => 1 = 0 + 1
001 => [3,1] => [3,1]
 => 1 = 0 + 1
011 => [2,1,1] => [2,1,1]
 => 1 = 0 + 1
111 => [1,1,1,1] => [1,1,1,1]
 => 0 = -1 + 1
0000 => [5] => [5]
 => 1 = 0 + 1
0001 => [4,1] => [4,1]
 => 1 = 0 + 1
0011 => [3,1,1] => [3,1,1]
 => 1 = 0 + 1
0111 => [2,1,1,1] => [2,1,1,1]
 => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 0 = -1 + 1
00000 => [6] => [6]
 => 1 = 0 + 1
00001 => [5,1] => [5,1]
 => 1 = 0 + 1
00011 => [4,1,1] => [4,1,1]
 => 1 = 0 + 1
00111 => [3,1,1,1] => [3,1,1,1]
 => 1 = 0 + 1
01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 1 = 0 + 1
11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 0 = -1 + 1
=> [1] => [1]
 => 0 = -1 + 1
Description
The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000875
(load all 4 compositions to match this statistic)
Mapping
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
Mp00094: Integer compositions ā€”to binary wordāŸ¶ Binary words
St000875: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1 = 0 + 1
1 => [1,1] => 11 => 0 = -1 + 1
00 => [3] => 100 => 1 = 0 + 1
01 => [2,1] => 101 => 1 = 0 + 1
11 => [1,1,1] => 111 => 0 = -1 + 1
000 => [4] => 1000 => 1 = 0 + 1
001 => [3,1] => 1001 => 1 = 0 + 1
011 => [2,1,1] => 1011 => 1 = 0 + 1
111 => [1,1,1,1] => 1111 => 0 = -1 + 1
0000 => [5] => 10000 => 1 = 0 + 1
0001 => [4,1] => 10001 => 1 = 0 + 1
0011 => [3,1,1] => 10011 => 1 = 0 + 1
0111 => [2,1,1,1] => 10111 => 1 = 0 + 1
1111 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
00000 => [6] => 100000 => 1 = 0 + 1
00001 => [5,1] => 100001 => 1 = 0 + 1
00011 => [4,1,1] => 100011 => 1 = 0 + 1
00111 => [3,1,1,1] => 100111 => 1 = 0 + 1
01111 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
11111 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
=> [1] => 1 => 0 = -1 + 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001280
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
Mp00040: Integer compositions ā€”to partitionāŸ¶ Integer partitions
St001280: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 1 = 0 + 1
1 => [1,1] => [1,1]
 => 0 = -1 + 1
00 => [3] => [3]
 => 1 = 0 + 1
01 => [2,1] => [2,1]
 => 1 = 0 + 1
11 => [1,1,1] => [1,1,1]
 => 0 = -1 + 1
000 => [4] => [4]
 => 1 = 0 + 1
001 => [3,1] => [3,1]
 => 1 = 0 + 1
011 => [2,1,1] => [2,1,1]
 => 1 = 0 + 1
111 => [1,1,1,1] => [1,1,1,1]
 => 0 = -1 + 1
0000 => [5] => [5]
 => 1 = 0 + 1
0001 => [4,1] => [4,1]
 => 1 = 0 + 1
0011 => [3,1,1] => [3,1,1]
 => 1 = 0 + 1
0111 => [2,1,1,1] => [2,1,1,1]
 => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 0 = -1 + 1
00000 => [6] => [6]
 => 1 = 0 + 1
00001 => [5,1] => [5,1]
 => 1 = 0 + 1
00011 => [4,1,1] => [4,1,1]
 => 1 = 0 + 1
00111 => [3,1,1,1] => [3,1,1,1]
 => 1 = 0 + 1
01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 1 = 0 + 1
11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 0 = -1 + 1
=> [1] => [1]
 => 0 = -1 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001421
(load all 4 compositions to match this statistic)
Mapping
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
Mp00094: Integer compositions ā€”to binary wordāŸ¶ Binary words
St001421: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1 = 0 + 1
1 => [1,1] => 11 => 0 = -1 + 1
00 => [3] => 100 => 1 = 0 + 1
01 => [2,1] => 101 => 1 = 0 + 1
11 => [1,1,1] => 111 => 0 = -1 + 1
000 => [4] => 1000 => 1 = 0 + 1
001 => [3,1] => 1001 => 1 = 0 + 1
011 => [2,1,1] => 1011 => 1 = 0 + 1
111 => [1,1,1,1] => 1111 => 0 = -1 + 1
0000 => [5] => 10000 => 1 = 0 + 1
0001 => [4,1] => 10001 => 1 = 0 + 1
0011 => [3,1,1] => 10011 => 1 = 0 + 1
0111 => [2,1,1,1] => 10111 => 1 = 0 + 1
1111 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
00000 => [6] => 100000 => 1 = 0 + 1
00001 => [5,1] => 100001 => 1 = 0 + 1
00011 => [4,1,1] => 100011 => 1 = 0 + 1
00111 => [3,1,1,1] => 100111 => 1 = 0 + 1
01111 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
11111 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
=> [1] => 1 => 0 = -1 + 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St001485
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
Mp00094: Integer compositions ā€”to binary wordāŸ¶ Binary words
St001485: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1 = 0 + 1
1 => [1,1] => 11 => 0 = -1 + 1
00 => [3] => 100 => 1 = 0 + 1
01 => [2,1] => 101 => 1 = 0 + 1
11 => [1,1,1] => 111 => 0 = -1 + 1
000 => [4] => 1000 => 1 = 0 + 1
001 => [3,1] => 1001 => 1 = 0 + 1
011 => [2,1,1] => 1011 => 1 = 0 + 1
111 => [1,1,1,1] => 1111 => 0 = -1 + 1
0000 => [5] => 10000 => 1 = 0 + 1
0001 => [4,1] => 10001 => 1 = 0 + 1
0011 => [3,1,1] => 10011 => 1 = 0 + 1
0111 => [2,1,1,1] => 10111 => 1 = 0 + 1
1111 => [1,1,1,1,1] => 11111 => 0 = -1 + 1
00000 => [6] => 100000 => 1 = 0 + 1
00001 => [5,1] => 100001 => 1 = 0 + 1
00011 => [4,1,1] => 100011 => 1 = 0 + 1
00111 => [3,1,1,1] => 100111 => 1 = 0 + 1
01111 => [2,1,1,1,1] => 101111 => 1 = 0 + 1
11111 => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
=> [1] => 1 => 0 = -1 + 1
Description
The modular major index of a binary word. This is [[St000290]] modulo the length of the word.
Matching statistic: St000381
(load all 9 compositions to match this statistic)
Mapping
Mp00269: Binary words ā€”flag zeros to zerosāŸ¶ Binary words
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
St000381: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 2 = 0 + 2
1 => 1 => [1,1] => 1 = -1 + 2
00 => 01 => [2,1] => 2 = 0 + 2
01 => 10 => [1,2] => 2 = 0 + 2
11 => 11 => [1,1,1] => 1 = -1 + 2
000 => 011 => [2,1,1] => 2 = 0 + 2
001 => 101 => [1,2,1] => 2 = 0 + 2
011 => 110 => [1,1,2] => 2 = 0 + 2
111 => 111 => [1,1,1,1] => 1 = -1 + 2
0000 => 0111 => [2,1,1,1] => 2 = 0 + 2
0001 => 1011 => [1,2,1,1] => 2 = 0 + 2
0011 => 1101 => [1,1,2,1] => 2 = 0 + 2
0111 => 1110 => [1,1,1,2] => 2 = 0 + 2
1111 => 1111 => [1,1,1,1,1] => 1 = -1 + 2
00000 => 01111 => [2,1,1,1,1] => 2 = 0 + 2
00001 => 10111 => [1,2,1,1,1] => 2 = 0 + 2
00011 => 11011 => [1,1,2,1,1] => 2 = 0 + 2
00111 => 11101 => [1,1,1,2,1] => 2 = 0 + 2
01111 => 11110 => [1,1,1,1,2] => 2 = 0 + 2
11111 => 11111 => [1,1,1,1,1,1] => 1 = -1 + 2
=> => [1] => 1 = -1 + 2
Description
The largest part of an integer composition.
Matching statistic: St000808
(load all 9 compositions to match this statistic)
Mapping
Mp00269: Binary words ā€”flag zeros to zerosāŸ¶ Binary words
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
St000808: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 2 = 0 + 2
1 => 1 => [1,1] => 1 = -1 + 2
00 => 01 => [2,1] => 2 = 0 + 2
01 => 10 => [1,2] => 2 = 0 + 2
11 => 11 => [1,1,1] => 1 = -1 + 2
000 => 011 => [2,1,1] => 2 = 0 + 2
001 => 101 => [1,2,1] => 2 = 0 + 2
011 => 110 => [1,1,2] => 2 = 0 + 2
111 => 111 => [1,1,1,1] => 1 = -1 + 2
0000 => 0111 => [2,1,1,1] => 2 = 0 + 2
0001 => 1011 => [1,2,1,1] => 2 = 0 + 2
0011 => 1101 => [1,1,2,1] => 2 = 0 + 2
0111 => 1110 => [1,1,1,2] => 2 = 0 + 2
1111 => 1111 => [1,1,1,1,1] => 1 = -1 + 2
00000 => 01111 => [2,1,1,1,1] => 2 = 0 + 2
00001 => 10111 => [1,2,1,1,1] => 2 = 0 + 2
00011 => 11011 => [1,1,2,1,1] => 2 = 0 + 2
00111 => 11101 => [1,1,1,2,1] => 2 = 0 + 2
01111 => 11110 => [1,1,1,1,2] => 2 = 0 + 2
11111 => 11111 => [1,1,1,1,1,1] => 1 = -1 + 2
=> => [1] => 1 = -1 + 2
Description
The number of up steps of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
Matching statistic: St001235
(load all 7 compositions to match this statistic)
Mapping
Mp00280: Binary words ā€”path rowmotionāŸ¶ Binary words
Mp00178: Binary words ā€”to compositionāŸ¶ Integer compositions
St001235: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => 2 = 0 + 2
1 => 0 => [2] => 1 = -1 + 2
00 => 01 => [2,1] => 2 = 0 + 2
01 => 10 => [1,2] => 2 = 0 + 2
11 => 00 => [3] => 1 = -1 + 2
000 => 001 => [3,1] => 2 = 0 + 2
001 => 010 => [2,2] => 2 = 0 + 2
011 => 100 => [1,3] => 2 = 0 + 2
111 => 000 => [4] => 1 = -1 + 2
0000 => 0001 => [4,1] => 2 = 0 + 2
0001 => 0010 => [3,2] => 2 = 0 + 2
0011 => 0100 => [2,3] => 2 = 0 + 2
0111 => 1000 => [1,4] => 2 = 0 + 2
1111 => 0000 => [5] => 1 = -1 + 2
00000 => 00001 => [5,1] => 2 = 0 + 2
00001 => 00010 => [4,2] => 2 = 0 + 2
00011 => 00100 => [3,3] => 2 = 0 + 2
00111 => 01000 => [2,4] => 2 = 0 + 2
01111 => 10000 => [1,5] => 2 = 0 + 2
11111 => 00000 => [6] => 1 = -1 + 2
=> => [1] => 1 = -1 + 2
Description
The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
The following 341 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001642The Prague dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000142The number of even parts of a partition. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000347The inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000390The number of runs of ones in a binary word. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000628The balance of a binary word. St000651The maximal size of a rise in a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000691The number of changes of a binary word. St000761The number of ascents in an integer composition. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001393The induced matching number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001874Lusztig's a-function for the symmetric group. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001928The number of non-overlapping descents in a permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000259The diameter of a connected graph. St000288The number of ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000325The width of the tree associated to a permutation. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000388The number of orbits of vertices of a graph under automorphisms. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000903The number of different parts of an integer composition. St000935The number of ordered refinements of an integer partition. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000753The Grundy value for the game of Kayles on a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000289The decimal representation of a binary word. St000297The number of leading ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000792The Grundy value for the game of ruler on a binary word. St000877The depth of the binary word interpreted as a path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001721The degree of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000354The number of recoils of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000653The last descent of a permutation. St000730The maximal arc length of a set partition. St000794The mak of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000919The number of maximal left branches of a binary tree. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001114The number of odd descents of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001592The maximal number of simple paths between any two different vertices of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000990The first ascent of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001352The number of internal nodes in the modular decomposition of a graph. St000260The radius of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000699The toughness times the least common multiple of 1,. St000023The number of inner peaks of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001500The global dimension of magnitude 1 Nakayama algebras. St001734The lettericity of a graph. St000806The semiperimeter of the associated bargraph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000661The number of rises of length 3 of a Dyck path. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in 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. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001586The number of odd parts smaller than the largest even part in 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. St001961The sum of the greatest common divisors of all pairs of parts. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001118The acyclic chromatic index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001281The normalized isoperimetric number of a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. 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. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000464The Schultz index of a connected graph. 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. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001651The Frankl number of a lattice. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000477The weight of a partition according to Alladi. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.