Xmas Summation 2016

Question

Given that $$\begin{aligned} f(1)&=\text{partridge in a pear tree}\\ f(2)&=\text{turtle doves}\\ f(3)&=\text{French hens}\\ f(4)&=\text{calling birds}\\ f(5)&=\text{golden rings}\\ f(6)&=\text{geese a-laying}\\ f(7)&=\text{swans a-swimming}\\ f(8)&=\text{maids a-milking}\\ f(9)&=\text{ladies dancing}\\ f(10)&=\text{lords a-leaping}\\ f(11)&=\text{pipers piping}\\ f(12)&=\text{drummers drumming} \end{aligned}$$ formulate the lyrics to the song The Twelve Days of Christmas in summation form for maximum compactness.

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Note

This question is posted under the tag "recreational mathematics" in the spirit of festive cheer.

Answer 1

$$\sum_{i = 1}^{12} \sum_{j = 1}^i jf(j)$$

I believe that gives you everything that you would end up with over the twelve days.

To get the lyrics, you would want

$\sum_{i = 1}^{12}$"On the $i$-th day of Christmas my true love gave to me" + $\left( \sum_{j = -i}^{-1} -j + f(-j) \right)$

Answer 2

Throughout the tune, we receive 12 partridges in pear trees, 22 turtle doves, 30 french hens, etc. So to total everything up, I think this sum is more compact: $$\sum_{n=1}^{12}(13-n)f(n)$$

Answer 3

Define $$g(n)=\text{On the }n\text{-th day of Christmas my true love gave to me}$$ and the operator $\circ$ as the concatenation of two text strings.

The lyrics of the song can then be represented as $$\color{red}{\boxed{\sum_{n=1}^{12} g(n)\circ \sum_{m=1}^n (n+1-m)\circ f(n+1-m)}}$$

Note: To avoid confusion, the operator $\circ$ has been defined and used.