Identifier
- St000863: Permutations ⟶ ℤ
Values
[1] => 1
[1,2] => 2
[2,1] => 2
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 3
[2,3,1] => 3
[3,1,2] => 2
[3,2,1] => 3
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 3
[2,1,3,4] => 4
[2,1,4,3] => 3
[2,3,1,4] => 4
[2,3,4,1] => 4
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 3
[3,2,1,4] => 4
[3,2,4,1] => 4
[3,4,1,2] => 3
[3,4,2,1] => 4
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 4
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 4
[1,2,5,3,4] => 4
[1,2,5,4,3] => 3
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 4
[1,3,4,5,2] => 4
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 4
[1,4,2,5,3] => 3
[1,4,3,2,5] => 4
[1,4,3,5,2] => 4
[1,4,5,2,3] => 3
[1,4,5,3,2] => 4
[1,5,2,3,4] => 4
[1,5,2,4,3] => 3
[1,5,3,2,4] => 3
[1,5,3,4,2] => 3
[1,5,4,2,3] => 3
[1,5,4,3,2] => 4
[2,1,3,4,5] => 5
[2,1,3,5,4] => 4
[2,1,4,3,5] => 4
[2,1,4,5,3] => 4
[2,1,5,3,4] => 4
[2,1,5,4,3] => 3
[2,3,1,4,5] => 5
[2,3,1,5,4] => 4
[2,3,4,1,5] => 5
[2,3,4,5,1] => 5
[2,3,5,1,4] => 4
[2,3,5,4,1] => 4
[2,4,1,3,5] => 4
[2,4,1,5,3] => 4
[2,4,3,1,5] => 4
[2,4,3,5,1] => 4
[2,4,5,1,3] => 4
[2,4,5,3,1] => 4
[2,5,1,3,4] => 4
[2,5,1,4,3] => 3
[2,5,3,1,4] => 4
[2,5,3,4,1] => 4
[2,5,4,1,3] => 3
[2,5,4,3,1] => 4
[3,1,2,4,5] => 4
[3,1,2,5,4] => 3
[3,1,4,2,5] => 4
[3,1,4,5,2] => 4
[3,1,5,2,4] => 3
[3,1,5,4,2] => 3
[3,2,1,4,5] => 5
[3,2,1,5,4] => 4
[3,2,4,1,5] => 5
[3,2,4,5,1] => 5
[3,2,5,1,4] => 4
[3,2,5,4,1] => 4
[3,4,1,2,5] => 4
[3,4,1,5,2] => 4
[3,4,2,1,5] => 5
[3,4,2,5,1] => 5
[3,4,5,1,2] => 4
[3,4,5,2,1] => 5
[3,5,1,2,4] => 3
[3,5,1,4,2] => 3
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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 $Q$.
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 $Q$.
References
[1] Sagan, B. E. Shifted tableaux, Schur $Q$-functions, and a conjecture of R. Stanley MathSciNet:0883894
[2] Worley, D. R. A THEORY OF SHIFTED YOUNG TABLEAUX MathSciNet:2941073
[3] Haiman, M. D. On mixed insertion, symmetry, and shifted Young tableaux MathSciNet:0989194
[2] Worley, D. R. A THEORY OF SHIFTED YOUNG TABLEAUX MathSciNet:2941073
[3] Haiman, M. D. On mixed insertion, symmetry, and shifted Young tableaux MathSciNet:0989194
Code
def statistic(pi):
return GrowthDiagram.rules.ShiftedShapes()(pi).P_chain()[-1][0]
Created
Jun 25, 2017 at 20:09 by Martin Rubey
Updated
Feb 09, 2021 at 14:57 by Martin Rubey
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