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Your data matches 313 different statistics following compositions of up to 3 maps.
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Matching statistic: St000160
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000160: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 0 + 1
[2]
=> 1 = 0 + 1
[1,1]
=> 2 = 1 + 1
[3]
=> 1 = 0 + 1
[2,1]
=> 1 = 0 + 1
[1,1,1]
=> 3 = 2 + 1
[4]
=> 1 = 0 + 1
[3,1]
=> 1 = 0 + 1
[2,2]
=> 2 = 1 + 1
[2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 4 = 3 + 1
[5]
=> 1 = 0 + 1
[4,1]
=> 1 = 0 + 1
[3,2]
=> 1 = 0 + 1
[3,1,1]
=> 2 = 1 + 1
[2,2,1]
=> 1 = 0 + 1
[2,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1]
=> 5 = 4 + 1
[6]
=> 1 = 0 + 1
[5,1]
=> 1 = 0 + 1
[4,2]
=> 1 = 0 + 1
[4,1,1]
=> 2 = 1 + 1
[3,3]
=> 2 = 1 + 1
[3,2,1]
=> 1 = 0 + 1
[3,1,1,1]
=> 3 = 2 + 1
[2,2,2]
=> 3 = 2 + 1
[2,2,1,1]
=> 2 = 1 + 1
[2,1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,1,1]
=> 6 = 5 + 1
[7]
=> 1 = 0 + 1
[6,1]
=> 1 = 0 + 1
[5,2]
=> 1 = 0 + 1
[5,1,1]
=> 2 = 1 + 1
[4,3]
=> 1 = 0 + 1
[4,2,1]
=> 1 = 0 + 1
[4,1,1,1]
=> 3 = 2 + 1
[3,3,1]
=> 1 = 0 + 1
[3,2,2]
=> 2 = 1 + 1
[3,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 4 = 3 + 1
[2,2,2,1]
=> 1 = 0 + 1
[2,2,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> 5 = 4 + 1
[1,1,1,1,1,1,1]
=> 7 = 6 + 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000326
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00280: Binary words āpath rowmotionā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00280: Binary words āpath rowmotionā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 11 => 1 = 0 + 1
[2]
=> 100 => 011 => 2 = 1 + 1
[1,1]
=> 110 => 111 => 1 = 0 + 1
[3]
=> 1000 => 0011 => 3 = 2 + 1
[2,1]
=> 1010 => 1101 => 1 = 0 + 1
[1,1,1]
=> 1110 => 1111 => 1 = 0 + 1
[4]
=> 10000 => 00011 => 4 = 3 + 1
[3,1]
=> 10010 => 01101 => 2 = 1 + 1
[2,2]
=> 1100 => 0111 => 2 = 1 + 1
[2,1,1]
=> 10110 => 11011 => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 11111 => 1 = 0 + 1
[5]
=> 100000 => 000011 => 5 = 4 + 1
[4,1]
=> 100010 => 001101 => 3 = 2 + 1
[3,2]
=> 10100 => 11001 => 1 = 0 + 1
[3,1,1]
=> 100110 => 011011 => 2 = 1 + 1
[2,2,1]
=> 11010 => 11101 => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 110111 => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 111111 => 1 = 0 + 1
[6]
=> 1000000 => 0000011 => 6 = 5 + 1
[5,1]
=> 1000010 => 0001101 => 4 = 3 + 1
[4,2]
=> 100100 => 011001 => 2 = 1 + 1
[4,1,1]
=> 1000110 => 0011011 => 3 = 2 + 1
[3,3]
=> 11000 => 00111 => 3 = 2 + 1
[3,2,1]
=> 101010 => 110101 => 1 = 0 + 1
[3,1,1,1]
=> 1001110 => 0110111 => 2 = 1 + 1
[2,2,2]
=> 11100 => 01111 => 2 = 1 + 1
[2,2,1,1]
=> 110110 => 111011 => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 1101111 => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 1111111 => 1 = 0 + 1
[7]
=> 10000000 => 00000011 => 7 = 6 + 1
[6,1]
=> 10000010 => 00001101 => 5 = 4 + 1
[5,2]
=> 1000100 => 0011001 => 3 = 2 + 1
[5,1,1]
=> 10000110 => 00011011 => 4 = 3 + 1
[4,3]
=> 101000 => 110001 => 1 = 0 + 1
[4,2,1]
=> 1001010 => 0110101 => 2 = 1 + 1
[4,1,1,1]
=> 10001110 => 00110111 => 3 = 2 + 1
[3,3,1]
=> 110010 => 011101 => 2 = 1 + 1
[3,2,2]
=> 101100 => 110011 => 1 = 0 + 1
[3,2,1,1]
=> 1010110 => 1101011 => 1 = 0 + 1
[3,1,1,1,1]
=> 10011110 => 01101111 => 2 = 1 + 1
[2,2,2,1]
=> 111010 => 111101 => 1 = 0 + 1
[2,2,1,1,1]
=> 1101110 => 1110111 => 1 = 0 + 1
[2,1,1,1,1,1]
=> 10111110 => 11011111 => 1 = 0 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111111 => 1 = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00280: Binary words āpath rowmotionā¶ Binary words
Mp00105: Binary words ācomplementā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00280: Binary words āpath rowmotionā¶ Binary words
Mp00105: Binary words ācomplementā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 11 => 00 => 0
[2]
=> 100 => 011 => 100 => 1
[1,1]
=> 110 => 111 => 000 => 0
[3]
=> 1000 => 0011 => 1100 => 2
[2,1]
=> 1010 => 1101 => 0010 => 0
[1,1,1]
=> 1110 => 1111 => 0000 => 0
[4]
=> 10000 => 00011 => 11100 => 3
[3,1]
=> 10010 => 01101 => 10010 => 1
[2,2]
=> 1100 => 0111 => 1000 => 1
[2,1,1]
=> 10110 => 11011 => 00100 => 0
[1,1,1,1]
=> 11110 => 11111 => 00000 => 0
[5]
=> 100000 => 000011 => 111100 => 4
[4,1]
=> 100010 => 001101 => 110010 => 2
[3,2]
=> 10100 => 11001 => 00110 => 0
[3,1,1]
=> 100110 => 011011 => 100100 => 1
[2,2,1]
=> 11010 => 11101 => 00010 => 0
[2,1,1,1]
=> 101110 => 110111 => 001000 => 0
[1,1,1,1,1]
=> 111110 => 111111 => 000000 => 0
[6]
=> 1000000 => 0000011 => 1111100 => 5
[5,1]
=> 1000010 => 0001101 => 1110010 => 3
[4,2]
=> 100100 => 011001 => 100110 => 1
[4,1,1]
=> 1000110 => 0011011 => 1100100 => 2
[3,3]
=> 11000 => 00111 => 11000 => 2
[3,2,1]
=> 101010 => 110101 => 001010 => 0
[3,1,1,1]
=> 1001110 => 0110111 => 1001000 => 1
[2,2,2]
=> 11100 => 01111 => 10000 => 1
[2,2,1,1]
=> 110110 => 111011 => 000100 => 0
[2,1,1,1,1]
=> 1011110 => 1101111 => 0010000 => 0
[1,1,1,1,1,1]
=> 1111110 => 1111111 => 0000000 => 0
[7]
=> 10000000 => 00000011 => 11111100 => 6
[6,1]
=> 10000010 => 00001101 => 11110010 => 4
[5,2]
=> 1000100 => 0011001 => 1100110 => 2
[5,1,1]
=> 10000110 => 00011011 => 11100100 => 3
[4,3]
=> 101000 => 110001 => 001110 => 0
[4,2,1]
=> 1001010 => 0110101 => 1001010 => 1
[4,1,1,1]
=> 10001110 => 00110111 => 11001000 => 2
[3,3,1]
=> 110010 => 011101 => 100010 => 1
[3,2,2]
=> 101100 => 110011 => 001100 => 0
[3,2,1,1]
=> 1010110 => 1101011 => 0010100 => 0
[3,1,1,1,1]
=> 10011110 => 01101111 => 10010000 => 1
[2,2,2,1]
=> 111010 => 111101 => 000010 => 0
[2,2,1,1,1]
=> 1101110 => 1110111 => 0001000 => 0
[2,1,1,1,1,1]
=> 10111110 => 11011111 => 00100000 => 0
[1,1,1,1,1,1,1]
=> 11111110 => 11111111 => 00000000 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000648
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St000648: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St000648: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => 5
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => 6
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,2,3] => 4
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,2,5,6,4] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,3] => 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => 0
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000731
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
St000731: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
St000731: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct 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]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 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]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => 5
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => 6
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => 4
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => 0
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St001189
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
St001189: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
St001189: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 3
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 0
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 4
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000011
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00327: Dyck paths āinverse Kreweras complementā¶ Dyck paths
Mp00124: Dyck paths āAdin-Bagno-Roichman transformationā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00327: Dyck paths āinverse Kreweras complementā¶ Dyck paths
Mp00124: Dyck paths āAdin-Bagno-Roichman transformationā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 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]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[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]
=> 3 = 2 + 1
[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]
=> 1 = 0 + 1
[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]
=> 1 = 0 + 1
[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 = 1 + 1
[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]
=> 2 = 1 + 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]
=> 4 = 3 + 1
[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]
=> 1 = 0 + 1
[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]
=> 1 = 0 + 1
[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]
=> 1 = 0 + 1
[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 = 1 + 1
[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 = 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 = 2 + 1
[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]
=> 5 = 4 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000382
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1 = 0 + 1
[2]
=> [[1,2]]
=> [2] => [1] => 1 = 0 + 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2 = 1 + 1
[3]
=> [[1,2,3]]
=> [3] => [1] => 1 = 0 + 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => [1,1] => 1 = 0 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3 = 2 + 1
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1 = 0 + 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => [1,1] => 1 = 0 + 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2 = 1 + 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1] => 2 = 1 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4 = 3 + 1
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1 = 0 + 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => [1,1] => 1 = 0 + 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => [1,1] => 1 = 0 + 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => [2,1] => 2 = 1 + 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [1,2] => 1 = 0 + 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => [3,1] => 3 = 2 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5 = 4 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1 = 0 + 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => [1,1] => 1 = 0 + 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => [1,1] => 1 = 0 + 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => [2,1] => 2 = 1 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2 = 1 + 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => [1,1,1] => 1 = 0 + 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => [3,1] => 3 = 2 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3 = 2 + 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => [2,2] => 2 = 1 + 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => [4,1] => 4 = 3 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6 = 5 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1 = 0 + 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => [1,1] => 1 = 0 + 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => [1,1] => 1 = 0 + 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => [2,1] => 2 = 1 + 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => [1,1] => 1 = 0 + 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => [1,1,1] => 1 = 0 + 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => [3,1] => 3 = 2 + 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => [1,2] => 1 = 0 + 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => [2,1] => 2 = 1 + 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => [2,1,1] => 2 = 1 + 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => [4,1] => 4 = 3 + 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => [1,3] => 1 = 0 + 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => [3,2] => 3 = 2 + 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => [5,1] => 5 = 4 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7 = 6 + 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1 = 0 + 1
[2]
=> [[1,2]]
=> [2] => [1] => 1 = 0 + 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2 = 1 + 1
[3]
=> [[1,2,3]]
=> [3] => [1] => 1 = 0 + 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => [1,1] => 1 = 0 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3 = 2 + 1
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1 = 0 + 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1 = 0 + 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2 = 1 + 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2 = 1 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4 = 3 + 1
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1 = 0 + 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1 = 0 + 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1] => 1 = 0 + 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2 = 1 + 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,1] => 1 = 0 + 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 3 = 2 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5 = 4 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1 = 0 + 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1 = 0 + 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1 = 0 + 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2 = 1 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2 = 1 + 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 3 = 2 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3 = 2 + 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 2 = 1 + 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 4 = 3 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6 = 5 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1 = 0 + 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1 = 0 + 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [1,1] => 1 = 0 + 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2 = 1 + 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [1,1] => 1 = 0 + 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [1,1,1] => 1 = 0 + 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 3 = 2 + 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [2,1] => 1 = 0 + 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [1,2] => 2 = 1 + 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,1,2] => 2 = 1 + 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 4 = 3 + 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [3,1] => 1 = 0 + 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [2,3] => 3 = 2 + 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,5] => 5 = 4 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7 = 6 + 1
Description
The last part of an integer composition.
Matching statistic: St000876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00317: Integer partitions āodd partsā¶ Binary words
Mp00224: Binary words ārunsortā¶ Binary words
Mp00104: Binary words āreverseā¶ Binary words
St000876: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00224: Binary words ārunsortā¶ Binary words
Mp00104: Binary words āreverseā¶ Binary words
St000876: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1 => 1 = 0 + 1
[2]
=> 0 => 0 => 0 => 1 = 0 + 1
[1,1]
=> 11 => 11 => 11 => 2 = 1 + 1
[3]
=> 1 => 1 => 1 => 1 = 0 + 1
[2,1]
=> 01 => 01 => 10 => 1 = 0 + 1
[1,1,1]
=> 111 => 111 => 111 => 3 = 2 + 1
[4]
=> 0 => 0 => 0 => 1 = 0 + 1
[3,1]
=> 11 => 11 => 11 => 2 = 1 + 1
[2,2]
=> 00 => 00 => 00 => 2 = 1 + 1
[2,1,1]
=> 011 => 011 => 110 => 1 = 0 + 1
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4 = 3 + 1
[5]
=> 1 => 1 => 1 => 1 = 0 + 1
[4,1]
=> 01 => 01 => 10 => 1 = 0 + 1
[3,2]
=> 10 => 01 => 10 => 1 = 0 + 1
[3,1,1]
=> 111 => 111 => 111 => 3 = 2 + 1
[2,2,1]
=> 001 => 001 => 100 => 1 = 0 + 1
[2,1,1,1]
=> 0111 => 0111 => 1110 => 2 = 1 + 1
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5 = 4 + 1
[6]
=> 0 => 0 => 0 => 1 = 0 + 1
[5,1]
=> 11 => 11 => 11 => 2 = 1 + 1
[4,2]
=> 00 => 00 => 00 => 2 = 1 + 1
[4,1,1]
=> 011 => 011 => 110 => 1 = 0 + 1
[3,3]
=> 11 => 11 => 11 => 2 = 1 + 1
[3,2,1]
=> 101 => 011 => 110 => 1 = 0 + 1
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4 = 3 + 1
[2,2,2]
=> 000 => 000 => 000 => 3 = 2 + 1
[2,2,1,1]
=> 0011 => 0011 => 1100 => 1 = 0 + 1
[2,1,1,1,1]
=> 01111 => 01111 => 11110 => 3 = 2 + 1
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6 = 5 + 1
[7]
=> 1 => 1 => 1 => 1 = 0 + 1
[6,1]
=> 01 => 01 => 10 => 1 = 0 + 1
[5,2]
=> 10 => 01 => 10 => 1 = 0 + 1
[5,1,1]
=> 111 => 111 => 111 => 3 = 2 + 1
[4,3]
=> 01 => 01 => 10 => 1 = 0 + 1
[4,2,1]
=> 001 => 001 => 100 => 1 = 0 + 1
[4,1,1,1]
=> 0111 => 0111 => 1110 => 2 = 1 + 1
[3,3,1]
=> 111 => 111 => 111 => 3 = 2 + 1
[3,2,2]
=> 100 => 001 => 100 => 1 = 0 + 1
[3,2,1,1]
=> 1011 => 0111 => 1110 => 2 = 1 + 1
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5 = 4 + 1
[2,2,2,1]
=> 0001 => 0001 => 1000 => 2 = 1 + 1
[2,2,1,1,1]
=> 00111 => 00111 => 11100 => 1 = 0 + 1
[2,1,1,1,1,1]
=> 011111 => 011111 => 111110 => 4 = 3 + 1
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7 = 6 + 1
Description
The number of factors in the Catalan decomposition 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 number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
The following 303 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000899The maximal number of repetitions of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St001733The number of weak left to right maxima of a Dyck path. St000237The number of small exceedances. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000058The order of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000439The position of the first down step of a Dyck path. St000873The aix statistic of a permutation. St000654The first descent of a permutation. St000717The number of ordinal summands of a poset. St000996The number of exclusive left-to-right maxima of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000740The last entry of a permutation. St000989The number of final rises of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000678The number of up steps after the last double rise of a Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000485The length of the longest cycle of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000338The number of pixed points of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000331The number of upper interactions of a Dyck path. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000155The number of exceedances (also excedences) of a permutation. St000352The Elizalde-Pak rank of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000990The first ascent of a permutation. St001050The number of terminal closers of a set partition. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000054The first entry of the permutation. St000203The number of external nodes of a binary tree. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000971The smallest closer of a set partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001346The number of parking functions that give the same permutation. St001530The depth of a Dyck path. St000756The sum of the positions of the left to right maxima of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001061The number of indices that are both descents and recoils of a permutation. St001657The number of twos in an integer partition. St000153The number of adjacent cycles of a permutation. St000910The number of maximal chains of minimal length in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000295The length of the border of a binary word. St000007The number of saliances of the permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. 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. St000461The rix statistic of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000738The first entry in the last row of a standard tableau. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000477The weight of a partition according to Alladi. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000010The length of the partition. St000012The area of a Dyck path. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000185The weighted size of a partition. St000228The size of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit 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. St000442The maximal area to the right of an up step of a Dyck path. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St000885The number of critical steps in the Catalan decomposition of a binary word. 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. 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. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001587Half of the largest even part of an integer partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001438The number of missing boxes of a skew partition. St001868The number of alignments of type NE of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001816Eigenvalues of the top-to-random operator acting on a simple module. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000934The 2-degree of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001498The normalised height of a Nakayama algebra with magnitude 1. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001730The number of times the path corresponding to a binary word crosses the base line. St001948The number of augmented double ascents of a permutation. St000765The number of weak records in an integer composition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001360The number of covering relations in Young's lattice below a partition. St001525The number of symmetric hooks on the diagonal of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St000406The number of occurrences of the pattern 3241 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000516The number of stretching pairs of a permutation. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001847The number of occurrences of the pattern 1432 in a permutation. St000456The monochromatic index of a connected graph. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000259The diameter of a connected graph. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000454The largest eigenvalue of a graph if it is integral. St001651The Frankl number of a lattice. St001330The hat guessing number of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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. St000260The radius of a connected graph. St000993The multiplicity of the largest part of an integer partition. St001556The number of inversions of the third entry of a permutation. St001645The pebbling number of a connected graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. St001568The smallest positive integer that does not appear twice in the 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. St001487The number of inner corners of a skew partition. St000091The descent variation of a composition. St000365The number of double ascents of a permutation. 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. St000709The number of occurrences of 14-2-3 or 14-3-2. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St000090The variation of a composition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001151The number of blocks with odd minimum. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000839The largest opener of a set partition. 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.
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