Generator

genfun

A generator in the discrete library, called 5 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

genfun is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

A product of 3 polynomials, and what its coefficients count. The coefficients of a product of small polynomials, with the combinations of choices that reach one marked total written out beneath it.

`show: "convolution"`

Multiplying two series is summing a diagonal. A multiplication table of the coefficients of two series, with one anti-diagonal marked: its sum is the coefficient of the corresponding power in the product.

`show: "recursion"`

1 / (1 − x − x²), and the objects its coefficients count. Two rows of coefficients: those of a generating function, and a direct count of the objects it is meant to count. The rows agree at every power.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the coefficient of x^0 is the number of combinations totalling 0 ×10
the coefficient of x^0 is the number of objects of size 0 ×9
the coefficient of x^0 is the sum along its diagonal ×7
and the marked coefficient counts the combinations written out beside it ×1
between two and four boxes, each offering between two and five whole-number values ×1
both series are short lists of small whole numbers ×1
every combination of choices is listed ×1
the marked coefficient is a small whole power ×1
the marked coefficient is inside the product ×1
the marked total is one the boxes can reach ×1
the sequence is growing, so the check has content ×1
the sequence is one the family knows ×1
the series inverted has a non-zero constant term ×1
the series is drawn to between 5 and 10 terms ×1
the view is one the family draws ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Discrete

A polynomial that counts

Hang a counting sequence on the powers of a variable and the two ways of combining choices — this and that, this or that — become multiplication and addition, so a recursion turns into an equation and the equation can be solved.

The whole library · What the figures prove