The Young tableau (plural, "tableaux") of a Ferrers diagram is obtained by placing the numbers 1, ..., in the boxes of the diagram. A "standard" Young tableau
is a Young tableau in which the numbers form an increasing sequence along each line
and along 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 number 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).
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 .
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L. "Standard Tableaux." Ch. 2, Exercise 26 in Advanced
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Netherlands: Reidel, pp. 125-126, 1974.Fulton, W. Young
Tableaux with Applications to Representation Theory and Geometry. New York:
Cambridge University Press, 1997.Kreweras, G. "Sur une class de
problèmes de dénombrement liés au treillis des partitions d'entiers."
Cahiers Buro6, 2-107, 1965.Kreweras, G. "Dénombrements
de chemins minimaux à sauts imposés." Comptes Rendus Acad.
Sci. Paris263, 1-3, 1966.Kreweras, G. "Sur une extension
du problème dir 'de Simon Newcomb.' " Comptes Rendus Acad. Sci. Paris263,
43-45, 1966.Kreweras, G. "Traitement simultané du 'problème
de Young' et du 'problème de Simon Newcomb.' " Cahiers Buro10,
23-31, 1967.Messiah, A. Appendix D in Quantum
Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, p. 1113,
1961-62.Ruskey, F. "Information on Permutations." http://www.theory.csc.uvic.ca/~cos/inf/perm/PermInfo.html#Tableau.Skiena,
S. "Young Tableaux." §2.3 in Implementing
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N. J. A. Sequence A000085/M1221
in "The On-Line Encyclopedia of Integer Sequences."Stanley,
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Young Tableaux." J. Symb. Comput.20, 731-735, 1995.
in the 
boxes of the diagram. A "standard" Young tableau
is a Young tableau in which the numbers form an increasing sequence along each line
and along 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 number 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
and 
.
This is the same as the number of permutation
involutions on 
elements (Skiena 1990, p. 32).
.
