2.3. p(n)x n = n=0 i= x = i x x 2 x 3 x..,?. p(n)x n = + x + 2 x x 3 + x + 7 x + x + n=0, n p(n) x n, ( ). p(n) (mother function)., x i = + xi +

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1 ( ) : ( ) n, n., = 2+2+,, = = = ,,,. ( composition.), λ = (2, 2, )... n (partition), λ = (λ, λ 2,..., λ r ), λ λ 2 λ r > 0, r λ i = n i=. r λ, l(λ)., r λ i = n i=, λ, λ., n P n, p(n), (number of partitions).,, (), (, ), (3, 2), (3,, ), (2, 2, ), (2,,, ), (,,,,, ) 7. (), (,,,, ). (,,,, ), (2, 2, ) 2 2, (2,,, ) 2 3..,. P = {,, 2, 2, 3 2, 32, 3 3, 2 3, 2 2 2, 2, }.2. n = 2, 3,, P n., n = 0, 0 ( ), p(0) =. p() =, p(2) = 2, p(3) = 3, p() =, p() = 7, p() =, , masaoishikawa@okayama-u.ac.jp

2 2.3. p(n)x n = n=0 i= x = i x x 2 x 3 x..,?. p(n)x n = + x + 2 x x 3 + x + 7 x + x + n=0, n p(n) x n, ( ). p(n) (mother function)., x i = + xi + x 2 i + x 3 i + x i + i. i= x i = ( + x + x2 + ){ + x 2 + (x 2 ) 2 + },....3, n = 7, 8, 9 p(n). 2,. 2.. (Young ) λ = (λ, λ 2,..., λ r ), λ, 2 λ 2,..., r λ r, (Young diagram) Ferrers Graph. (cell), i i (ith row), j j (jth column), (Alfred Young) ,.,,,, 89, (, ) 0..,. 90,, 90., 902,, (Algebra of Invariants). 907,, ( ). 908,, 90 2 Birdbrook., 92.,,.

3 *** ( ) *** 3 i j (i, j) (cell (i, j)). (row number), (column number)., λ Y λ,, λ, λ., (i, j) Y λ (i, j) λ.., λ = λ Y λ = Y 2 3,,, 2, 3, 3,,. Y X X, 2, (2, ), Y, 3, 2 (3, 2) λ = (λ, λ 2,..., λ r ), Y λ (, ) λ (conjugate), λ = (λ,..., λ s)., s λ., 7 λ = 2 3 Y λ = Y 2 3, λ = Y , λ = (,,, 3,, ) ( ) λ = (λ, λ 2,..., λ r ) Y λ, (i, j) Y λ (i, j). (i, j) ( ), ( ) (i, j) H λ (i, j), (i, j) (hook),, (i, j)., (hook), (i, j) (hook length), h λ (i, j). (i, j) x = (i, j), H λ (x), h λ (x). x a a l l

4 , λ = 2 3 Y 2 3 x = (2, 2), x a ( (arm) ), x l ( (leg) ), x. H λ (x) = H λ (2, 2) = {(2, 2), (2, 3), (2, ), (3, 2), (, 2)}, x = (2, 2), H λ (x) h λ (x) = h λ (2, 2) =.?, λ = 2 3 Y 2 3, λ Y λ, h λ (x) = x, 2 (corner)., x = (i, j) Y λ, (i, j + ), (i +, j) Y λ., λ = 2 3 Y λ Y λ, (, ), (3, ), (, 3) h λ (i, j) = λ i j + λ j i + (2.). x a a a a l l l, λ = 2 3 Y 2 3 x = (, 2), λ 2 = 2 =, λ 2 = = 3, h 2 3(, 2) = = 8.. Y λ (i, j) λ i j, λ j i, (i, j). 2.

5 *** ( ) *** 3,,., Alfred Young,,,,,,,. 3.. ( ) λ = (λ, λ 2,..., λ r ) n, Y λ. Y λ n., Y λ,..., n, T i) ii), T (standard Young tableau),, (standard tableau). 3, λ, (shape)., λ S λ, SYT λ., , 7,,.,,?,?,., 3 2 S 3 2,, SYT 3 2 = (Frame-Robinson-Thrall []) λ n ( λ = n), λ, n! SYT λ x λ h (3.) λ(x) 3 tableau,. tableaux..

. Frame-Robinson -Thrall, (hook length formula).,, 3 2,.!,, 3 2 Y 3 2 3 2 3 2.,. n! x λ h λ(x) = 3 2 3 3 2 2 =,., 32, Y 32 3 3., n! x λ h λ(x) = 3 2 3 3 = Gilbert de Beauregard Robinson (90 992) 90.

6 . Frame-Robinson -Thrall, (hook length formula).,, 3 2,.!,, 3 2 Y ,. n! x λ h λ(x) = =,., 32, Y , n! x λ h λ(x) = = Gilbert de Beauregard Robinson (90 992) 90. St. Andrew, 927.,,,.,, 97,,,. 938,, Robinson-Schensted,. 0,, (The Foundations of Geometry) (90 ), (The Representations of the Symmetric Groups) (9 ), (Vector Geometry) ,,,,.,,, 9.,, H (Best Kept Secret)., (SIGINT Examination Unit).,,, (Most Excellent Order of the British Empire)., 9 (Canadian Mathematical Congress), H.S.M., (Canadian Journal of Mathematics). 99,, 30, (Managing Editor) , (Canadian Mathematical Society) (president), 99.,,, ( ),, (the Society for the History and Philosophy of Mathematics), NRC (National Research Council)

7 *** ( ) *** 7., SYT 32 =,.,,, Greene-Nijenhuis-Wilf [2]... λ SYT λ SYT λ = µ SYT µ (.).,, Y λ, µ., λ = 2 3 Y 2 3, c c 3 c 2 c = (, ), c 2 = (3, ), c 3 = (, 3) 3. c, c 2, c 3, 3 µ.. λ n, T S λ, λ, n,., n, µ T., λ µ T,, n, λ T.?, λ = 32, c = (, 3), c 2 = (2, 2), c 3 = (3, ) 3. c = (, 3) T

8 8. c µ = 2 2 T., c = (, 3), T., c 2 = (2, 2) T , c 2 = (2, 2) µ = 3 2 T., c = (2, 2), T., c 3 = (3, ) T c µ = 32 T., c 3 = (3, ), T., c, c 2, c 3 µ = 2 2, 3 2, 32, λ = 32 SYT 32 = SYT SYT SYT 32 = + + =,. (.)., (.), (3.), (.),., (3.),..2. λ n, F λ = n! x λ h λ(x). λ =, F λ =., x = (α, β) λ, λ, x µ, F µ F λ (α, β)., F λ = (α,β) F λ (α, β) (.2)

9 *** ( ) *** 9.,,, Y λ (α, β).,, λ = λ =,,. (.2) F λ = (α,β) F λ (α, β) F λ (.3).,, (.3)., (.3),... ( ) λ n.., Y λ (i, j )., Y λ /n. (n = λ )., (i, j ), (i, j ) (i 2, j 2 )., /(h λ (i, j ) )., (i 2, j 2 ) (i 2, j 2 ),., /(h λ (i 2, j 2 ) )., Y λ,.,., (α, β) p λ (α, β)., λ = 32, p 32 (, 3). x x = (, 3), (, ), (, 2), (, 3),., (, ), 2 (, 2) (, 3) x, (, 2),., (, 2), (, 3) (2, 2) (, 3). (, 3) 2,., (, ) Start (, 2) (, 3)., p 32 (2, 2) (, 2) (, 3) (, 3) (, 3) p 32 (, 3) = = x

10 0, (, ) (2, ) Start (, 2) (, 2) (2, ) (2, 2) (2, 2) (2, 2) (2, 2) (2, 2). p 32 (2, 2) = = p 32 (3, )., p λ (α, β)?,,.,,,..3. λ n, (α, β) Y λ.,. p λ (α, β) = F λ(α, β) F λ (.), (.), (.3).,.,, (α, β),!.,.3, Frame-Robinson- Thrall.,,,.3.. (α, β), c = (α, β) µ, h µ (i, j), c (i, β), i < α, c (α, j), j < β h µ (i, j) = h λ (i, j),, h µ (i, j) = h λ (i, j), (.). x x x x x c F λ (α, β) F λ = n = n α i= α i= β h λ (i, β) h λ (i, β) j= ( + h λ (i, β) h λ (α, j) h λ (α, j) ) β j= ( ) + h λ (α, j)., h λ (α, β) =.,,.

11 *** ( ) ***.. ( ) λ, (A, B). A = {a,..., a k }, B = {b,..., b l } a < a 2 < < a k, b < b 2 < < b l. k, l., (a k, b l ) λ Y λ., (a, b ), (a, b ), (a k, b l ),,, A, B E A,B., (a, b ) = (i, j ) (i 2, j 2 ) (i m, j m ) = (a k, b l ) {i,..., i m } = A {j,..., j m } = B., (a, b ) E A,B p λ (A, B a, b )., (a, b ), /n., λ = 2 3 Y 2 3 A = {, 3}, B = {, 2, }, c = (i, j ) c 2 = (i 2, j 2 ) c 3 = (i 3, j 3 ) c = (i, j ) {i, i 2, i 3, i } = A {j, j 2, j 3, j } = B, 3. c c 2 c 3 c c c 2 c 3 c c c 2 c 3 c,. p 2 3 (A, B, ) = = 8, λ = 32, A = {2}, B = {, 2}., (2, ) (2, 2), Start (2, ) (2, 2)., p 32 ({2}, {, 2} 2, ) = 2

12 2. (2, ),., A = {, 2}, B = {, 2}, (, ) (2, 2), A = {, 2}, B = {, 2}, Start (, ) (, 2) (2, ) (2, 2) (2, 2),... p 32 ({, 2}, {, 2}, ) = = p 32 ({, 2}, {2}, 2) = 2 p 32 ({2}, {2} 2, 2) =... λ, (A, B) = ({a,..., a k }, {b,..., b l }), (a k, b l ) Y λ. p λ (A, B a, b ) = k i= l h λ (a i, b l ) j= h λ (a k, b j ) (.2).,.., (a, b ),, (a, b 2 ),, (a 2, b ), h λ (a, b )., (a, b 2 ),, b, (a, b 2 ), (A, {b 2,..., b l }),, (a 2, b ),, a, (a 2, b ), ({a 2,..., a k }, B). p λ (A, B a, b ) = h λ (a, b ) p λ (A, B \ {b } a, b 2 ) + h λ (a, b ) p λ (A \ {a }, B a 2, b ) { = pλ (A, B \ {b } a, b 2 ) + p λ (A \ {a }, B a 2, b ) } (.3) h λ (a, b )., B \ {b } B b {b 2,..., b l }. A \ {a } {a 2,..., a k }.,, B \ {b } = A \ {a } =, p λ (A, a, b 2 ) = p λ (, B a 2, b ) = 0., (.3), (.2) A + B. A A.

13 *** ( ) *** 3 (i) A + B = 2, A, B A = {a }, B = {b },, (.2), (a, b ),,. (ii) n 3, A + B = n, (.2). (.3) p λ (A, B a, b ) = { k h λ (a, b ) i= + l h λ (a i, b l ) j=2 k i=2 = h λ(a, b l ) + h λ (a k, b ) h λ (a, b ) l h λ (a i, b l ) j= k i= h λ (a k, b j ) h λ (a k, b j ) l h λ (a i, b l ) j= } h λ (a k, j)., 2. (2.) h λ (a, b l ) = λ a + λ b l a b l +, h λ (a k, b ) = λ ak + λ b a k b +, h λ (a, b ) = λ a + λ b a b +, h λ (a k, b l ) = λ ak + λ b l a k b l + h λ (a, b l ) + h λ (a k, b ) = h λ (a, b ) + h λ (a k, b l ), (a k, b l ), h λ (a k, b l ) = p λ (A, B a, b ) = k i= l h λ (a i, b l ) j=., (.2), A + B = n. (i), (ii), (.2), (A, B). h λ (a k, j),,., λ = 2 3 A = {, 3}, B = {, 2, }, start x x x end, (.2) x h λ (, ) h λ (3, ) h λ (3, 2) = 3 = 8., p λ (A, B, ) = 8.,.3 (.3). (.3) F λ (α, β) F λ = n α i= ( + h λ (i, β) ) β j= ( ) + h λ (α, j)

14 ,. α i= ( + ) = h λ (i, β) h A a A λ (a, β), h λ. (i,β) i A = {a,..., a k } {,..., α }., {,..., α} A. a k = α A. α i= ( + ) = h λ (i, β) A., A = {a,..., a k } a A a α a < < a k = α h λ (a, β)., β j= ( +, B = {b,..., b l } ) = h λ (α, j) B b B b β b < < b l = β h λ (α, b)., F λ (α, β) F λ = n k (A,B) i= l h λ (a i, β) j= h λ (α, b j ).,. (.2), F λ (α, β) F λ = n p λ (A, B a, b ) (A,B)., (a, b ) /n, (A, B), (α, β), p λ (α, β).,.3 (.).,,,., λ = 32, (α, β) = (2, 2) x (.3) F λ (2, 2) = ( + F λ ) ( + h λ (, 2) ) h λ (2, )

15 *** ( ) ***, F λ (2, 2) F λ = ( + h λ (, 2) + h λ (2, ) + h λ (, 2).. (.2). F λ (2, 2) F λ p λ ({2}, {2} 2, 2) = p λ ({, 2}, {2}, 2) = h λ (, 2) = 2 p λ ({2}, {, 2}, 2) = h λ (2, ) = 2 p λ ({, 2}, {, 2}, ) = h λ (, 2) h λ (2, ) = ) h λ (2, ) = {p λ ({2}, {2} 2, 2) + p λ ({, 2}, {2}, 2) + p λ ({2}, {, 2}, 2) + p λ ({, 2}, {, 2}, )} = ( ) = 3 8 (2, 2) p λ (2, 2).,,.,,, ( ).,,,.,,,., ( ), 920 (Georg Ferdinand Frobenius ) Charlottenburg. Christian Ferdinand Frobenius Christine Elizabeth Friedrich. 80 Joachimsthal Gymnasium, 87,,., (Kronecker), (Kummer), (Karl Weierstrass), 870.,,,. Sophienrealschule Joachimsthal Gymnasium, 87,,, (Eidgenossische Polytechnikum). 7,.,. 89 2,,.,,,,. 893,., Prussian Academy of Sciences.,... (Issaj Schur, Issai Schur, ),. ( ) ( )., ,, 99.

, Lie, Lie,.,,,,,,,,. 7 [] J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric group. Canad. J. Math. (9), 3 32.

16 , Lie, Lie,.,,,,,,,,. 7 [] J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric group. Canad. J. Math. (9), [2] Curtis Greene, Albert Nijenhuis, and Herbert S Wilf, A probabilistic proof of a formula for the number of Young tableaux of a given shape, Advances in Mathematics, 3 (979), [3] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs (999). [] Richard P. Stanley, Enumerative Combinatorics: Volume (Cambridge Studies in Advanced Mathematics), Cambridge University Press (202). [] Richard P. Stanley, Enumerative Combinatorics: Volume 2 (Cambridge Studies in Advanced Mathematics), Cambridge University Press (200). [], ( ), (2002). [7],, (2009). [8] Alfred Young. Quantitative substitutional analysis II, Proc. London Math. Sot., Ser., 3 (902), ,,., 93., 93,, ,.., ( ),.,,,,,.. 929,. 7 Acknowledgement:,., [7],.

λ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T

λ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T 0 2 8 8 6 3 0 0 Young Young [F] 0.. Young λ n λ n λ = (λ,, λ l ) λ λ 2 λ l λ = ( m, 2 m 2, ) λ = n, l(λ) = l {λ n n 0} P λ = (λ, ), µ = (µ, ) n λ µ k k k λ i µ i λ µ λ = µ k i= i= i < k λ i = µ i λ k >