Mathematica Asked by user75105 on December 7, 2020
I am given a set like some terms from the partition of $20$:
${{17, 3}, {13, 7}, {11, 3, 3, 3}, {7, 7, 3, 3}, {7, 5, 5, 3}, {5, 5,
5, 5}}$.
From the above data,
$20= underline{1}*17+underline{1}*3= underline{1}*13+underline{1}*7 =underline{1}*11+underline{3}*3 = underline{2}*7+underline{2}*3 = underline{1}*7+underline{2}*5+underline{1}*3 = underline{4}*5$
Now the above underlined numbers indicate the number of repetitions of each term appearing in the partition. I need to write those number of repetitions in the form of summation involving factorials:
Like for the above case, I need to represent the summation for the above partition as:
${T=frac{1}{1!*1!}+frac{1}{1!*1!}+frac{1}{1!*3!}+frac{1}{2!*2!}+frac{1}{1!*2!}+frac{1}{4!}}$
How can I do that?
fn=Tr[1/Times @@@ (Tally[#][[All, 2]] & /@ #!)]&;
fn@{{17, 3}, {13, 7}, {11, 3, 3, 3}, {7, 7, 3, 3}, {7, 5, 5, 3}, {5, 5,
5, 5}}
71/24
Answered by ciao on December 7, 2020
ip = {{17, 3}, {13, 7}, {11, 3, 3, 3}, {7, 7, 3, 3}, {7, 5, 5, 3}, {5, 5, 5, 5}};
Total[Times @@ Values[1 / Factorial @ #] & /@ Counts /@ ip]
71/24
Also
Query[Total, Times @@ Values[1/Factorial @ #] &] @ Dataset[Counts /@ ip]
71/24
Dataset[Counts /@ ip][Total, Times @@ Values[1/Factorial @ #] &]
71/24
Answered by kglr on December 7, 2020
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