Identifier
Values
[1,0] => [1,0] => 10 => 11 => 1
[1,0,1,0] => [1,1,0,0] => 1100 => 1001 => 1
[1,1,0,0] => [1,0,1,0] => 1010 => 1111 => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 101101 => 1
[1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => 100001 => 1
[1,1,0,0,1,0] => [1,0,1,0,1,0] => 101010 => 111111 => 1
[1,1,0,1,0,0] => [1,0,1,1,0,0] => 101100 => 111001 => 1
[1,1,1,0,0,0] => [1,1,0,0,1,0] => 110010 => 100111 => 1
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 10100101 => 1
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 10111101 => 1
[1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => 11010100 => 10000001 => 1
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 10001101 => 1
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 10110001 => 1
[1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0] => 10110010 => 11100111 => 1
[1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 11111111 => 1
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => 10110100 => 11100001 => 1
[1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 11101101 => 1
[1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => 11111001 => 1
[1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0] => 11010010 => 10000111 => 1
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 10011111 => 1
[1,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0] => 11001100 => 10011001 => 1
[1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 10110111 => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 1110101000 => 1011111101 => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 101010010101 => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 111110010000 => 101011000101 => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => 101110000101 => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => 101001100101 => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 101110010001 => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 101100110001 => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 111010010100 => 101111000001 => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 101100100101 => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 101001110001 => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 111110000100 => 101011010001 => 1
[1,1,0,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 111010010001 => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 111000110001 => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => 111011000001 => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 111010000101 => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 111000100101 => 1
[1,1,0,1,1,0,0,1,0,0,1,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 111110000001 => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 111000011001 => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 111001100001 => 1
[1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 100110011001 => 1
[1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 100110110001 => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 100111100001 => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 100110100101 => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 101100011001 => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 101101100001 => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => 101001011001 => 1
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searching the database for statistics with the same generating function
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.
Map
Knuth-Krattenthaler
Description
The map that sends the Dyck path to a 321-avoiding permutation, then applies the Robinson-Schensted correspondence and finally interprets the first row of the insertion tableau and the second row of the recording tableau as up steps.
Interpreting a pair of two-row standard tableaux of the same shape as a Dyck path is explained by Knuth in [1, pp. 60].
Krattenthaler's bijection between Dyck paths and $321$-avoiding permutations used is Mp00119to 321-avoiding permutation (Krattenthaler), see [2].
This is the inverse of the map Mp00127left-to-right-maxima to Dyck path that interprets the left-to-right maxima of the permutation obtained from Mp00024to 321-avoiding permutation as a Dyck path.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
alternating inverse
Description
Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.