# [seqfan] Falling factorials divided by rising factorials

jnthn stdhr jstdhr at gmail.com
Sun Feb 20 20:26:10 CET 2022

```Dear seqfans,

Please forgive the length of this post, but I am a bit peeved.

I am looking for help naming a sequence. This is my second attempt at
finding a concise and understandable name.

This sequence derives from an irregular triangle, where each row is
built by taking integer results from divisions of falling factorials by
rising factorials.

The first row begins with n=2, because 2! is the first factorial that
allows for a non-empty numerator (the falling factorial) and a non-empty
denominator (the rising factorial); {[2], [1]}, and [2]/[1]=2,

The first seven rows are:

2;

6, 3;

24, 12, 6, 4, 2;

120, 60, 30, 20, 10, 5;

720, 360, 180, 120, 60, 20, 30, 15, 5, 6, 3, 1;

5040, 2520, 1260, 840, 420, 140, 210, 105, 35, 42, 21, 7, 7;

40320, 20160, 10080, 6720, 3360, 1120, 1680, 840, 280, 70, 336, 168, 56,
14, 56, 28, 8, 4;

Here is an example of of the fractions being considered, with the rising
and falling factorials presented as sets, rather than products. For n=5,

n, falling factorial/rising factorial = x

5, [5, 4, 3, 2] / [1] = 120
5, [5, 4, 3] / [1] = 60
5, [5, 4, 3] / [2, 1] = 30
5, [5, 4] / [1] = 20
5, [5, 4] / [2, 1] = 10
5, [5, 4] / [3, 2, 1] = 3.333333
5, [5] / [1] = 5
5, [5] / [2, 1] = 2.5
5, [5] / [3, 2, 1] = 0.833333
5, [5] / [4, 3, 2, 1] = 0.208333,

which is why row four of the triangle contains 6 integers.

The motivation for this sequence arose from the question: when one
removes a central sequential product from n!, and then divides the
remaining falling factorial by the remaining rising factorial, when do we
get an integer result?

Using the notation (n)_k for the falling factorial (as suggested by an
editor), and the standard notation n! for the rising factorial, the
elements of each row are determined by starting with the maximal falling
factorial (i.e., k=2), (n)_2=n*(n-1)*...*2, and then dividing that number
by all rising factorials i! in increasing order, with i ranging from 1 to
(k-1), and updating the row with each integer result. Then the next smaller
falling factorial, (n)_3, is then divided by all i! where 0<i<3. And so on,
until we end by testing the fractions: (n)_(n)=n / 1!; n / 2!; ...; n /
(n-1)!

To (hopefully) further clarify was is going on, I'll point out that, when
viewed as a set of integers, the cardinality of the sequencial product
being removed ranges from 0 to n-2, so that we always have a valid fraction
to test -- e.g in the example with n=5 above, the cardinality of the
central sequential product removed from 5! in the first fraction is zero,
while the cardinality of the central sequential product removed in the
seventh fraction is n-2.

My first attempt at finding a good name was scuttled by an editor,
because the name that was provided by another well-intentioned editor was,
apparently, too confusing. The ordering of the row elements was different
in that attempt, but I don't believe that is relevant.

They asked, "Please can you sort the terms in each row in ascending
order. Otherwise you need to properly define the ordering and explain why
the ordering you have chosen is a natural choice."

Questions:
(1) Does asking to sort the terms make sense?

(2). Is there a "natural choice" of ordering of the row elements?
(3) Is there a *concise* and straight forward way to explain why the
elements of the rows will never be well-ordered (see last row of example
triangle)?

This indicated to me that the editor didn't understand what was going on.

Further, they stated "you seem to be removing duplicates and sometimes
not."

I provided an example similar to n=5 above, and responed that they seemed
to not understand what was going on, and that I didn't understand what they
meant by "duplicates", so could they please explain, using my example, what
they meant by "duplicates." Their response was: " I'm not sure it is for me
to be understanding what is going on. Suggest to reject. Poorly prepared
nonsense."

How can one edit , or meaningfully comment on, a sequence that they don't
feel the need to understand?

Is this sequence nonsense?

Moving on, we can change the order of the row elements in various ways.
For example, we could begin by dividing the least falling factorial (n)_(n)
by all i!, and proceeding on up to the greatest falling factorial (n)_2. In
doing so, we produce a (somewhat) different sequence, where the values
found in each row never change, just their order. Is there a "natural
choice" of ordering these row elements, as implied by the editor?

Since reordering the row elements changes the sequence, would there be
any reason to add any/all of these other sequences?

If this sequence, or the family of sequences, is worthy of submission, I
will allocated a block of A-numbers based on comments to this post.

Apologies for the rant.

Sincerely,
-jnthn

P.S.

For anyone familiar with python:

def fallingFactorialList(n,k):
f=[]
for k in range(n,k,-1):
f.append(k)
return f

def risingFactorialList(n):
f=[]
for k in range(n,0,-1):
f.append(k)
return f

def prod(lst):
product=1
for d in lst:
product*=d
return product

tri=[]
for n in range(2,11):
row=[]
risingFactorials=[]
for j in range(1,n):
risingFactorials.append(risingFactorialList(j))
fallingFactorials=[]
for k in range(1,n):
fallingFactorials.append(fallingFactorialList(n,k))
for numerator in fallingFactorials:
for denominator in risingFactorials:
if numerator[-1]==denominator[0]:
break
fraction=prod(numerator)/prod(denominator)
print(n, fraction, numerator, denominator)
if fraction.is_integer():
row.append(int(fraction))
tri.append(row)
seq=[]
for r in tri:
print(r)
for m in r:
seq.append(m)
print(seq)

```