- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000863: Permutations ⟶ ℤ

Values

0 => [2] => [1,1,0,0] => [1,2] => 2
1 => [1,1] => [1,0,1,0] => [2,1] => 2
00 => [3] => [1,1,1,0,0,0] => [1,2,3] => 3
01 => [2,1] => [1,1,0,0,1,0] => [1,3,2] => 2
10 => [1,2] => [1,0,1,1,0,0] => [2,1,3] => 3
11 => [1,1,1] => [1,0,1,0,1,0] => [2,3,1] => 3
000 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 4
001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 3
100 => [1,3] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 4
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 3
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 4
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 5
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 4
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 4
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 4
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 4
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 4
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 4
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 5
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 4
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 5
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 4
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 5
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 6
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => 5
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => 5
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => 5
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => 5
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => 4
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,2,4,5,3,6] => 5
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => 5
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => 5
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => 4
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => 4
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => 4
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,4,2,5,6] => 5
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => 4
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,4,5,2,6] => 5
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => 5
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => 6
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => 5
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 5
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => 5
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => 4
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => 5
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 5
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 6
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => 5
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => 5
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => 5
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 6
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => 5
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => 6
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 6
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 7
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,2,3,4,5,7,6] => 6
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,2,3,4,6,5,7] => 6
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [1,2,3,4,6,7,5] => 6
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,2,3,5,4,6,7] => 6
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,2,3,5,4,7,6] => 5
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,2,3,5,6,4,7] => 6
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [1,2,3,5,6,7,4] => 6
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,2,4,3,5,6,7] => 6
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,2,4,3,5,7,6] => 5
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5,7] => 5
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,2,4,3,6,7,5] => 5
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,2,4,5,3,6,7] => 6
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,2,4,5,3,7,6] => 5
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,2,4,5,6,3,7] => 6
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,2,4,5,6,7,3] => 6
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,3,2,4,5,6,7] => 6
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,3,2,4,5,7,6] => 5
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2,4,6,5,7] => 5
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,3,2,4,6,7,5] => 5
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,5,4,6,7] => 5
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,7,6] => 4
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,5,6,4,7] => 5
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,5,6,7,4] => 5
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,3,4,2,5,6,7] => 6
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,3,4,2,5,7,6] => 5
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,3,4,2,6,5,7] => 5
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,3,4,2,6,7,5] => 5
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,3,4,5,2,6,7] => 6
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,3,4,5,2,7,6] => 5
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,3,4,5,6,2,7] => 6
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,7,2] => 6
 => [1] => [1,0] => [1] => 1

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$F_{1} = 2\ q^{2}$

$F_{2} = q^{2} + 3\ q^{3}$

$F_{3} = 4\ q^{3} + 4\ q^{4}$

$F_{4} = q^{3} + 10\ q^{4} + 5\ q^{5}$

$F_{5} = 6\ q^{4} + 20\ q^{5} + 6\ q^{6}$

Description

The length of the first row of the shifted shape of a permutation.
The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled.
This statistic records the length of the first row of

$P$ and

$Q$ .

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.