Open Access Library Journal
Vol.07 No.12(2020), Article ID:106353,15 pages
10.4236/oalib.1107040
Subset of the Shape of Numbers
Ji Peng
Department of Electronic Information, Nanjing University, Nanjing, China
Copyright © 2020 by author(s) and Open Access Library Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: November 26, 2020; Accepted: December 28, 2020; Published: December 31, 2020
ABSTRACT
This article is based on the concept of Shape of numbers, introduces subset of the Shape and obtains its calculation formula. This article also makes some analysis and draws new conclusions, especially the calculation method of
. The Shape’s concept becomes clearer and richness.
Subject Areas:
Discrete Mathematics
Keywords:
Shape of Numbers, Calculation Formula, Sum of Powers of Integers, Combinatorics, Congruence, Stirling Number
1. Introduction
Peng, J. has introduced Shape of numbers in [1] [2]:
There are M-1 intervals between adjacent numbers.
means continuity,
means discontinuity.
Shape of numbers: collect
with the same continuity and discontinuity at the same position into a catalog, call it a Shape. A Shape has a min Item:
, Use the symbol PS = [min Item] to represent it. If
then only
is allowed.
The single
is an item,
is the product. Ii is a factor.
Example:
Define:
SET(N, PS) = set of items belonging to PS in
PM(PS) = count of factors
PB(PS) = count of discontinuities
MIN(PS) = min product:
,
IDX(PS) = (max factor)+1
PH(PS) = IDX(PS) − PB(PS) − 2
Basic Shape: intervals = 1 or 2
BASE(PS) = BS: if 1) PB(BS) = PB(PS), 2) PM(BS) = PM(PS), 3) BS is a Basic Shape 4) BS has discontinuity intervals at the same positions of PS.
Example:
|SET(N, PS)| = Count of items in SET(N, PS)
SUM(N, PS) = Sum of all products in SET(N, PS)
Example:
[1] [2] came to the following conclusion:
1.1)
1.2)
, PS is a Basic Shape
The following uses count of
for count of
1.3)
,
Use the form
, Xi = Gi or Ki.
The expansion has 2M items, don’t swap the factors of
, then each
corresponds to one expression =
, q = count of
, 2M items in total.
,
Example:
,
?
, K is fixed, E is variable.
, or
That is, a product can be broken down into 2M parts.
Define
= Sum of one part of
PF indicates the part.
, or
Rewrite 1.3) and add {braces}:
Let
expand by the {braces}:
1.4)
with same
,
Example:
, ?
Expand by the {braces}:
?
This can explain why 1.3) has that strange form:
We can calculate every part of 1.3) by some way without 1.3). There may be complex relationships between the parts, but their sum just match a simple form.
1.5) Use the symbol of 1.3), when
,,
Sum traverses all (N1, N2)-Choice of K
This ? 1.3) is compatible with 1.2)
1.6) P is a prime number, {PS1, PS2, …} are all of the Basic Shapes,
,,,
That is, them are Basic shapes, have same count of factors and same count of discontinuities > 0, and max factor = P − 1, then
Example:
2. Subset of SET(N, PS)
,,
If PB(PS) = 0,
is very simple.
If PB(PS) > 0, some changes appear in SET(N, PS).
We can fix some discontinuities of the Shape to get subsets.
Define SET(N, PS, PT) =Subset of SET(N, PS), a valid
(*)
others are invalid.
Example:
,, is invalid.
,
,
,
PT only has the change at (*). When a change happens, make the interval fixed.
The more changes, the fewer items:
Define PCHG(PS, PT) = count of change from BASE(PS) to PT
Example:
, changed at T1
, changed at T2
, changed at T1, T2
2.1)
2.2)
If PT1 only change Ti of PT, Obvious:
2.3) If PT1 only change Ti of PT,
.
Let
, then
In particular:
?
[Proof]
PT1 change Ti of PT ?
?
?
Count of the Items is equal.
Every item in SET(N, PS1, PT) is in SET(N, PS, PT), and not in SET(N, PS, PT1).
q.e.d.
if
, PT changes at Ti and Tj, then
Let
,
,
?
General:
2.4) The relationship between SET(N, PS, PT) and SET(N, PSX) is similar to the Inclusion Exclusion Principle.
3. Calculation formula of SET(N, PS, PT)
Define:
SUM_SUBSET(N, PS, PT) = Sum of all products in SET(N, PS, PT)
When PT is invalid, SUM_SUBSET(N, PS, PT) = 0
Only valid PT is discussed below.
,,
3.1) Use the form
, then
,
[Proof]
1) If PT = BS, then
? the formula holds.
2) If M = 1 and PT has 1 change, then
,,,
Let
, 2.3) →
The form = (T1 + K1) ? The formula holds.
3) If M > 1 and PT only has 1 change at TM, then
Let
, 2.3) →
:
When
,
Use the symbol of (1.3)
,
Let
,
? Match of the form
4) If M > 1 and PT only has 1 change at
, Let
, use the same method of (3).
5) if
, Use 2.3) → divide the Items into subset ? deducing by induction.
q.e.d.
Example:
?
Among:
;
Use the same method of 3.1)
3.2) Calculation formula of SUM_K(SET(N, PS, PT), PF) is similar to 1.4).
Example:
?
3.3) Use the form
,
,, 2M-1 Items in total.
In particular:
If
, then
?
3.4)
4. Analysis of SUM_SUBSET(N, PS, [1, 2, ∙∙∙, M + 1])
,. The simplest subset of PS is SET(N, PS, PT1).
(1*)
Solve (1*) in a normal way:
Decompose
to
?
,
4.1)
, 3.1) can decompose
to
In particular, 1.5) ?
?
?
4.2)
4.3) P is a prime number,
1)
,
2)
3) if
and
, then
[Proof]
and
q.e.d.
When
and
?
can be slided to
by MOD P
If
and exists
, PS can be slided to
If
and not exists
, PS must be a Basic Shape, can only be slided to
Define:
A Basic Shape
, among:
Li = count of continuity. (Li, Li+1) means a discontinuity, there are Q-1 discontinuities.
Example:
,,
Obvious:
can been slided to
,,
4.4) PS is a Basic Shape,
,, are all shapes that PS can scroll to, then
. This is a promotion of 1.6)
[Proof]
If
,
…
q.e.d.
Example:
5. Calculation Formula of
Use the form
.
In general, Ki and Kj cannot be exchanged, but when
,
Easy to see:
Can prove:
is permutable in
?
is permutable in the form.
Add one more factor Ki to the end:
Let
,
,
means
, means
It’s match the form
Recursion ?
5.1)
,, are allowed,
can use the form
Example:
The form =
?
Among:
The form =
?
Among:
In particular:
5.2)
Example:
The form =
?
The form =
?
The form =
?
Among:
S(M, K) is Stirling number of the second kind,
Definition of S(M, K) ?
It’s equal to 5.2), so we have a way to calculate S(M, K).
3.4) can be seen as
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Cite this paper
Peng, J. (2020) Subset of the Shape of Numbers. Open Access Library Journal, 7: e7040. https://doi.org/10.4236/oalib.1107040
References