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challset3sol2

# challset3sol2 - Back to MAT 344 Tucker 6.3 Selected...

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Back to MAT 344 Tucker 6.3: Selected Solutions 3. Find a generating function for the number of ways to write the integer r as a sum of positive integers in which no integer appears more than three times. Solution. This is the same as the number of integer solutions to 1*e 1 + 2*e 2 + 3*e 3 + ... r*e r with the conditions 0 <= e i <= 3 for all i. The generating function that models this is (1+x+x 2 +x 3 ) * (1+x 2 +x 4 +x 6 ) * (1+x 3 +x 6 +x 9 ) * ... (1+x k +x 2k +x 3k ) * ... 7. (a) Show that the number of partitions of 10 into distinct parts (integers) is equal to the number of partitions of 10 into odd parts by listing partitions of these two types. Solution. Distinct: 10, 1+9, 2+8, 3+7, 4+6, 1+4+5, 1+3+6, 1+2+7, 2+3+5, 1+2+3+4. Odd: 1+1+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+3, 1+1+1+1+3+3, 1+3+3+3, 1+1+3+5, 1+1+1+1+1+5, 5+5, 1+1+1+7, 3+7, 1+9. 7. (b) Show algebraically that the generating function for partitions of r into distinct parts equals the generating function for partitions of r into odd parts, and hence the numbers of these two types of partitions are equal.

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challset3sol2 - Back to MAT 344 Tucker 6.3 Selected...

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