New identities in Combinatorics discovered with Mathematica

Eric Dolores-Cuenca
Yonsei University

(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf

ABSTRACT

In our study of nonlinear Signal-flow graphs we used the software Mathematica to obtain evidence about new combinatorial identities. Eventually we proved those identities.

Polychrony

While I was working in AI, we had meetings with groups from neuroscience.

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\videos\\tope.gif", "AnimatedImage"]

Cellular automata 192

The previous diagram satisfies the condition of cellular automata 192, when we are allowed to correct outputs at different times.

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\rule192.png"]

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\videos\\tope.gif", "AnimatedImage"]

Social media / nonlinear signal-flow graphs

Why do we retweet a post?
External stimuli:
	A bot: you are paid to retweet whatever content. 
Internal stimuli:
	-Agreement,
          -viewer apathy (you follow the others),
          -image management (what others should see on my profile),     
          -indirect messages (I put it on the air  but I aim someone),
Non linearity: I retweet one time if one or more of these conditions hold.

Polychrony

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\videos\\tope.gif", "AnimatedImage"]

Polychrony

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\videos\\middlee.gif", "AnimatedImage"]

Left: external activations. Right: polycrhony groups.

Polychrony

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\triangulos.png"]

We stack a triangle into a second triangle. We do not allow the top vertices to coincide.

Chinampas

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\standart.png"]

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\chinampa.jpg"]

A black vertex represents a neuron activated either by internal or external stimuli.
Combinatorics + topology = algebraic topology

Chinampas

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\videos\\ks.gif", "AnimatedImage"]

n = # roots +1
k = height

Chinampas

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\videos\\k1.gif", "AnimatedImage"]

n = 3
k = 1 
with generating function

Chinampas

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\videos\\k3p.gif", "AnimatedImage"]

n = 3
k = 3
each diagram is determined by a solution to the system:
 with  and .
First configuration:
1<2<3<4<5
  2<3<4
Last configuration:
3<4<5<6<7
  4<5<6<7

Evidence

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\eexperiments0.png"]

Evidence

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\eexperiments1.png"]

Evidence

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\eexperiments2.png"]

Evidence

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\eexperiments3.png"]

Evidence

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\eexperiments.png"]

Experiments with Wolfram alpha

Given the line with  points, we build the graph .
Can we transform the generating function of the line into the generating function of ?

Experiments with Wolfram alpha

Experiments with Wolfram alpha

-use symbolic computing to determine  , where  is a polynomial in  n. All possible ways to put labels on  with numbers from  to n,

Experiments with Wolfram alpha

-use symbolic computing to determine  , where  is a polynomial in  n. All possible ways to put labels on  with numbers from  to n,
 	-ignore the coefficient  ,

Experiments with Wolfram alpha

-use symbolic computing to determine  , where  is a polynomial in  n. All possible ways to put labels on  with numbers from  to n,
 	-ignore the coefficient  ,
	-replace any instance of  by  ,   by  , etc. This defines the differential operator .

Example

f[x_] := x^13/(1-x)^14

Sum[1,{a,1,n-10}, {b,a+1,n-9}, {c,b+1,n-8}, {d,c+1,n-7}, {e,d+1,n-6}, {f,e+1,n-5},{g,f+1,n-4}, {h,g+1 ,n-3},{i,h+1,n-2},{j,i+1,n-1},{k,j+1,n}]

FullSimplify[(1/39916800)(3628800 n-10628640 (n^2)+12753576 (n^3)-8409500 (n^4)+3416930 (n^5)-902055 (n^6)+157773 (n^7)-18150 (n^8)+1320 (n^9)-55 (n^10)+(n^11))]

graph=

Sum[(C-a-1)*(E2-C-1)*(G2-E2-1)*(I-G2-1)*(k-I-1)*(g-e-1),{a,1,n-10},{C,a+2,n-8},{E2,C+2,n-6},{G2,E2+2,n-4},{I,G2+2,n-2},{k,I+2,n},{b,a+1,C-1},{D,C+1,E2-1},{F2,E2+1,G2-1},{H,G2+1,I-1},{j,I+1,k-1},{c,b+1,D-1},{E1,D+1,F2-1},{G1,F2+1,H-1},{i,H+1,j-1},{d,c+1,E1-1},{F1,E1+1,G1-1},{h,G1+1,i-1},{e,d+1,F1-1},{g,F1+1,h-1}]

FullSimplify[%5]

%6/%3

Expand[(-8+n) (-7+n) (((-6+n))^2) (((-5+n))^3) (((-4+n))^2) (-3+n) (-2+n) (6369+(-10+n) n (395+6 (-10+n) n))]

WolframAlphaQueryParseResults

WolframAlphaQueryParseResults

(-341575*x*D[%18, x])/2615348736000

%10+%11+%12+%13+%14+%15+%16+%17+%19+%20+%21+%22

FullSimplify[%23]

Expand[(x^11) (1+x) (1+x (98+x (3031+x (41708+x (295111+x (1155650+x (2598191+x (3401660+x (2598191+x (1155650+x (295111+x (41708+x (3031+x (98+x))))))))))))))]

Experiments

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\eexperiments.png"]

Time to do math

Since we know that symmetries were encrypted in the coefficients of the power series, we rediscovered generating functions of order polynomials. But with our modern point of view, we were able to find the following identities:

Time to do math

Time to do math

Let 
If

Open problems

Import["C:\\Users\\FSU\\Documents\\Python Scripts\\posets\\img\\eexperiments.png"]

Open problems

always has a odd digit.
 ,

References:

Power series representing Posets
 José Antonio Arciniega Nevárez, Eric Rubiel Dolores-Cuenca

Polychrony as Chinampas
Eric Dolores-Cuenca, José Antonio Arciniega-Nevárez, Anh Nguyen, Yitong Zou, Nathan Crock, Gordon Erlebacher, José L. Mendoza-Cortes