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E. g. 2 ~X [3,2,1 ]=; X /~ yields X X X X [(3,2,1 2 )']= X X =[4,2,1]. X Partitions a and Young diagrams [a] where a = a' are called self-associated. 4 P(n): = {yly~n} of all the partitIOns of n. ;;;; is a total order, so that the order diagram is always linear. 6 In the case when this holds we say that f3 dominates a and call ~ the dominance order. ) It is easy to see that the dominance order differs from the lexicographic order on pen) if and only if n~6. 8 a~f3 = a';;;;f3. J{3: [a

We can apply this if we happen to know the number of such matrices. 18 LEMMA. If r l , rz , c I , and C z are nonnegative integers with the property r l +rz =c I +c z ' then the number of2X2 matrices over No with row sums r l , rz and column sums c l ' Cz is equal to 1+min{r p rz , c l ' cz}. 2 The Permutation Characters Induced by Young Subgroups Proof If for example r\ =min{r" r2 , c" c2 }, then we have the following I + r I choices for the entries of the first row of such a 2 X 2 matrix: (: r\ -r) * ' Each of these choices yields exactly one 2 X 2 matrix with row sums r, and column sums c" since r] was assumed to be the minimum, so that a suitable second row can be found and is uniquely determined.

4 P(n): = {yly~n} of all the partitIOns of n. ;;;; is a total order, so that the order diagram is always linear. 6 In the case when this holds we say that f3 dominates a and call ~ the dominance order. ) It is easy to see that the dominance order differs from the lexicographic order on pen) if and only if n~6. 8 a~f3 = a';;;;f3. J{3: [a

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[Article] Insights on bias and information in group-level studies by Sheppard L.


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