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Identifier

St000179: Integer partitions ⟶ ℤ

Values

[] => 1

[1] => 1

[2] => 2

[1,1] => 2

[3] => 6

[2,1] => 3

[1,1,1] => 6

[4] => 24

[3,1] => 8

[2,2] => 12

[2,1,1] => 8

[1,1,1,1] => 24

[5] => 120

[4,1] => 30

[3,2] => 24

[3,1,1] => 20

[2,2,1] => 24

[2,1,1,1] => 30

[1,1,1,1,1] => 120

[6] => 720

[5,1] => 144

[4,2] => 80

[4,1,1] => 72

[3,3] => 144

[3,2,1] => 45

[3,1,1,1] => 72

[2,2,2] => 144

[2,2,1,1] => 80

[2,1,1,1,1] => 144

[1,1,1,1,1,1] => 720

[7] => 5040

[6,1] => 840

[5,2] => 360

[5,1,1] => 336

[4,3] => 360

[4,2,1] => 144

[4,1,1,1] => 252

[3,3,1] => 240

[3,2,2] => 240

[3,2,1,1] => 144

[3,1,1,1,1] => 336

[2,2,2,1] => 360

[2,2,1,1,1] => 360

[2,1,1,1,1,1] => 840

[1,1,1,1,1,1,1] => 5040

[8] => 40320

[7,1] => 5760

[6,2] => 2016

[6,1,1] => 1920

[5,3] => 1440

[5,2,1] => 630

[5,1,1,1] => 1152

[4,4] => 2880

[4,3,1] => 576

[4,2,2] => 720

[4,2,1,1] => 448

[4,1,1,1,1] => 1152

[3,3,2] => 960

[3,3,1,1] => 720

[3,2,2,1] => 576

[3,2,1,1,1] => 630

[3,1,1,1,1,1] => 1920

[2,2,2,2] => 2880

[2,2,2,1,1] => 1440

[2,2,1,1,1,1] => 2016

[2,1,1,1,1,1,1] => 5760

[1,1,1,1,1,1,1,1] => 40320

[9] => 362880

[8,1] => 45360

[7,2] => 13440

[7,1,1] => 12960

[6,3] => 7560

[6,2,1] => 3456

[6,1,1,1] => 6480

[5,4] => 8640

[5,3,1] => 2240

[5,2,2] => 3024

[5,2,1,1] => 1920

[5,1,1,1,1] => 5184

[4,4,1] => 4320

[4,3,2] => 2160

[4,3,1,1] => 1680

[4,2,2,1] => 1680

[4,2,1,1,1] => 1920

[4,1,1,1,1,1] => 6480

[3,3,3] => 8640

[3,3,2,1] => 2160

[3,3,1,1,1] => 3024

[3,2,2,2] => 4320

[3,2,2,1,1] => 2240

[3,2,1,1,1,1] => 3456

[3,1,1,1,1,1,1] => 12960

[2,2,2,2,1] => 8640

[2,2,2,1,1,1] => 7560

[2,2,1,1,1,1,1] => 13440

[2,1,1,1,1,1,1,1] => 45360

[1,1,1,1,1,1,1,1,1] => 362880

[10] => 3628800

[9,1] => 403200

[8,2] => 103680

[8,1,1] => 100800

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Description

The product of the hook lengths of the integer partition.
Consider the Ferrers diagram associated with the integer partition. For each cell in the diagram, drawn using the English convention, consider its hook: the cell itself, all cells in the same row to the right and all cells in the same column below. The hook length of a cell is the number of cells in the hook of a cell. This statistic is the product of the hook lengths of all cells in the partition.
Let $H_\lambda$ denote this product, then the number of standard Young tableaux of shape $\lambda$, (traditionally denoted $f^\lambda$) equals $n! / H_\lambda$. Therefore, it is consistent to set the product of the hook lengths of the empty partition equal to $1$.

References

[1] pp.472 Stanley, R. P. Enumerative combinatorics. Vol. 2 MathSciNet:1676282

Code

def statistic(L):
    return prod( L.hook_length(*cell) for cell in L.cells() )

#CodeLanguage: Mathematica
Needs["Combinatorica`"];
HookProduct[] := Module[{ips, stat, SageForm},
   SageForm[part_] := 
    StringJoin["[", Riffle[ToString /@ part, ","], "]"];
   ips = Join @@ (IntegerPartitions /@ Range[17]);
   stat[p_] := Tr[p]!/Combinatorica`NumberOfTableaux[p];
   Print[StringJoin @@ 
     Table[StringJoin[SageForm[p], "=>", ToString[stat[p]], "\n"], {p,
        ips[[1 ;; 1200]]}]];
   ];
HookProduct[]

Created

Mar 31, 2014 at 15:51 by Per Alexandersson

Updated

Oct 29, 2017 at 16:25 by Martin Rubey