A Young tableau (plural, "tableaux") is a filling of the boxes of a Ferrers diagram with positive
integers . A standard Young tableau with bumping
algorithm is used to construct a standard Young tableau from a permutation
of A000085 ). These numbers can be generated
by the recurrence relation
for integers permutation
involutions on
The number of all possible standard Young tableaux of a given shape can also be considered, and can be calculated with the hook length formula .
For example, the illustration above shows the 35 standard tableaux of shape
The Schensted correspondence is a bijection between permutations
of
A semistandard Young tableau has entries weakly increasing along rows and strictly increasing down columns. Entries may repeat.
See also Bumping Algorithm ,
Durfee Square ,
Ferrers Diagram ,
Hook
Length Formula ,
Partition ,
Partition
Function P ,
Permutation Involution ,
Random Tableau ,
Schensted
Correspondence ,
Tableau Class ,
Tableau
Promotion
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References Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture Was Solved." Not. Amer. Math. Soc. 46 , 637-646,
1999. Comtet, L. "Standard Tableaux." Ch. 2, Exercise
26 in Advanced
Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Fulton, W. Young
Tableaux: With Applications to Representation Theory and Geometry. Kreweras, G. "Sur une classe de
problèmes de dénombrement liés au treillis des partitions des
entiers." Cahiers Buro 6 , 9-107, 1965. Kreweras, G.
"Dénombrements de chemins minimaux à sauts imposés."
C. R. Acad. Sci. Paris 263 , 1-3, 1966. Kreweras, G.
"Sur une extension du problème dir 'de Simon Newcomb.' " C. R.
Acad. Sci. Paris 263 , 43-45, 1966. Kreweras, G. "Traitement
simultané du 'problème de Young' et du 'problème de Simon Newcomb.'
" Cahiers Buro 10 , 23-31, 1967. Messiah, A. Appendix
D in Quantum
Mechanics, Vol. 2. Ruskey, F. "Information on Permutations." https://web.archive.org/web/20140923074938/http://www.theory.csc.uvic.ca/~cos/inf/perm/PermInfo.html#Tableau . Skiena,
S. "Young Tableaux." §2.3 in Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Skiena, S. S. The
Algorithm Design Manual. Sloane,
N. J. A. Sequence A000085 /M1221
in "The On-Line Encyclopedia of Integer Sequences." Stanley,
R. P. Enumerative
Combinatorics, Vol. 1. Wilf, H. "The Computer-Aided Discovery of a Theorem about
Young Tableaux." J. Symb. Comput. 20 , 731-735, 1995. Referenced
on Wolfram|Alpha Young Tableau
Cite this as:
Weisstein, Eric W. "Young Tableau." From
MathWorld https://mathworld.wolfram.com/YoungTableau.html
Subject classifications
boxes contains each of 1, 2, ..., 
once, with entries increasing along each row and down each
column. For example, the standard Young tableaux of size 
are given by 
, 
, 
, and 
, illustrated above. The bumping
algorithm is used to construct a standard Young tableau from a permutation
of 
, and the numbers of standard
Young tableaux of size 1, 2, 3, ... are 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496,
... (OEIS A000085). These numbers can be generated
by the recurrence relation
, with 
and 
. This is the same as the number of permutation
involutions on 
elements (Skiena 1990, p. 32).
.
elements and pairs of standard Young
tableaux of the same shape, each containing 1, 2, ..., 
once.
