Partitions

A partition of a positive integer $k$ is a way of writing $k$ as a sum of positive integers listed in non-increasing order. We write a partition as a sequence $(a_1, a_2, \dots, a_n)$ with $a_1 \ge a_2 \ge \dots \ge a_n \ge 1$ and $a_1 + a_2 + \dots + a_n = k$. For example, $(12)$, $(2,2,2,2,2,2)$, and $(5,3,2,1,1)$ are all partitions of $12$.

Given two distinct partitions $A = (a_1, a_2, \dots, a_n)$ and $B = (b_1, b_2, \dots, b_m)$, let $t$ be the first position at which they differ (that is, $a_i = b_i$ for every $i < t$ and $a_t \ne b_t$). We say $A > B$ when $a_t > b_t$. Because $A$ and $B$ are partitions of the same integer, such a position $t$ always exists.

This rule orders all partitions of $k$ lexicographically, from the smallest to the largest. For example, the partitions of $5$ in order are:

(1,1,1,1,1)
(2,1,1,1)
(2,2,1)
(3,1,1)
(3,2)
(4,1)
(5)

Given $k$ and a positive integer $a$, find the $a$-th partition (counting from $1$) in this ordered list of partitions of $k$.

The first line contains $N$, the number of test cases. Each of the next $N$ lines contains two positive integers $k$ and $a$.

For each test case, output the $a$-th partition of $k$ in lexicographic order, written as its parts separated by commas and enclosed in parentheses, for example (5,3,2,1,1). If $a$ is greater than the total number of partitions of $k$, output Too big instead.