Young diagrams and Young tableaux

A Young diagram is an arrangement of boxes that satisfies the following conditions.

• The boxes are contiguous in every row and in every column.
• Every row is aligned to the leftmost column.
• No row is longer than the row directly above it.

An arrangement whose row lengths from top to bottom are 3, 2, 2, 1 is a Young diagram.

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A Young tableau fills the boxes of a Young diagram with numbers under the following conditions.

• Each box holds an integer from 1 to $N$, both ends allowed.
• The integer in a box is greater than or equal to the integer in the box directly to its left.
• The integer in a box is greater than the integer in the box directly above it.

With $N = 3$ and row lengths 3, 2, 1, the arrangement below satisfies all three conditions.

1 1 2
2 3
3

Given $N$ and the shape of a Young diagram, write a program that computes the number of ways to build a Young tableau.

The input holds several test cases and continues to the end of the file. Each test case takes two lines.

The first line gives the shape of the Young diagram. The leading integer $k$ is the number of rows, with $1 \le k \le 7$. Then come $k$ integers $l_1, l_2, \dots, l_k$, the box count of each row, satisfying $7 \ge l_1 \ge l_2 \ge \dots \ge l_k \ge 1$.

The second line gives $N$, with $k \le N \le 7$.

For each test case, print the number of Young tableaux that can be built, one per line.