Numerical Exploration of the Compacted Associated Stirling Numbers Khaled Ben Letaïef To cite this version: Khaled Ben Letaïef. Numerical Exploration of the Compacted Associated Stirling Numbers. 2017. <hal-01578309> HAL Id: hal-01578309 https://hal.archives-ouvertes.fr/hal-01578309 Submitted on 29 Aug 2017 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Notes on Number Theory and Discrete Mathematics ISSN 1310 5132 Vol. XX, XXXX, No. X, XX XX Numerical Exploration of the Compacted Associated Stirling Numbers Khaled Ben Letaïef 1 1 Aeronautics and aerospace high graduate engineer 16 Bd du Maréchal de Lattre, apt. 095, 21300 Chenove, France e-mail : letaiev@gmail.com Abstract : In this work, we make some Python simulations of the modular properties of associated Stirling numbers in order to illustrate our previous results and to make new conjectures. Keywords : Number Theory, Associated Stirling Numbers, Modular Rotations, Numerical Simulations. AMS Classification : 11B73, 05A18. Table des matières 1 The computer program in Python 2 2 Variation of n at r fixed 4 3 Variation of r at n fixed 7 4 Conclusion 14 Introduction The associated Stirling numbers of first and second kind at order r, respectively noted {d r (n, k)} n,k N and {s r (n, k)} n,k N, verify the following relations [4] : 1
d r (n, k) = (n 1)d r (n 1, k) + (n 1) r 1 d r (n r, k 1) ( ) n 1 s r (n, k) = k.s r (n 1, k) + s r (n r, k 1) r 1 In [1], we defined σ r and π r as the arithmetical triangles, respectively {D r (m, k)} m,k N and {S r (m, k)} m,k N, obtained from the application (r 1) times of the following linear geometrical transformation R (defined in the space of numerical sequences s with two integer variables n et k) to the associated Stirling numbers of first and second kind at order r : Then, by definition : R : s(n, k) R(s(n, k)) = s(n + k 1, k) with m = n (r 1)(k 1). D r (m, k) = R r 1 (d r )(n (r 1)(k 1), k) = d r (n, k) S r (m, k) = R r 1 (s r )(n (r 1)(k 1), k) = s r (n, k) This elastic transformation R was assimilated to a rotation, compacting the associated Stirling numbers from their usual dilated staircase structures into arithmetical triangles t verifying the general relation : t(i, j) = g(i, j)t(i 1, j 1) + h(i, j)t(i 1, j) Then, in [2] [3], modular properties in geometrical form have been obtained for σ r and π r and their corresponding associated Stirling numbers. In the current work, our purpose will be more experimental as we are going to numerically simulate some of those interesting and fundamentally visual properties but also try to infer new ones. 1 The computer program in Python Here, we use a Python code written by us for the simulation of π r modulo n for different values of r and n. It will show the modular behaviour we already proved in [1] [2] [3] when r varies. As for the notations, the 0 are the elements of π r divisible by n, whereas the 1 (or sometimes * ) are non divisible by n. The main geometrical lines defined in our theorems are put in red colour. Those examples visually show the modular rotations with increasing r. We have privileged Python language in our programming because of its power to handle very large integers. The following code, which provides LaTeX files of π r, was implemented under Cygwin. It can be easily generalized to the simulation of σ r. 2
# Program of calculation modulo n of the triangle pi_r (or sigma_r) r=input( order r= ) n=input( modulo n= ) h = input( number of lines of the triangle= ) # initialization of the triangle T=[[1,0]] # Loop for calculating the triangle as a list for i in range(1,h): T.append([1]) # initialization for T[i][0]=1 for j in range(1,i+1): prod=1 fact=1 for k in range(1,r): prod=(i+(r-1)*j+ k)*prod fact=k*fact # calculating the factorial of r # use of the recurrence relation of the triangle: T[i].append(prod//fact*T[i-1][j-1] + (j+1)*t[i-1][j]) T[i].append(0) # initialization for T[i][i+1]=0 # one can use this program for calculating sigma_r: # just do not calculate fact=r! and replace the top line # with: T[i].append(prod*T[i-1][j-1] + (j+1)*t[i-1][j]). # Computation in loop modulo n of the terms of the triangle # the 0 are the terms divisible by n # the 1 are the others for i in range(0,h): for j in range(0,i+1): if T[i][j]%n==0: # terms divisible by n are represented in blue "0" (red "0" on the diagonal): if i+(r-1)*j==n-r: # diagonal associated with n T[i][j]= \\textcolor{ + str( red ) + }{ + str( 0 ) + } else: T[i][j]= \\textcolor{ + str( blue ) + }{ + str( 0 ) + } else: # terms non divisible by n symbolized by green "1" (red "1" on the diagonal): T[i][j]= \\textcolor{ + str( green ) + }{ + str( 1 ) + } # red asymptote associated with n while its terms are not divisible by n: if i-j==n-r and T[i][j]== \\textcolor{ +str( green )+ }{ +str( 1 )+ } : 3
T[i][j]= \\textcolor{ + str( red ) + }{ + str( 1 ) + } # the centre of rotation of the line associated with n may be noted "*": T[n-r][0]= \\textcolor{ + str( red ) + }{ + str( * ) + } f=open( r= +str(r)+, + n= +str(n)+.tex, w ) # opening of the recording file of the results: f.write("$") # writing of the text in LaTeX and of the computation constants: f.write("\pi") f.write("_") f.write("{") f.write(str(r)) f.write("}") f.write("$") f.write("\n") f.write("modulo ") f.write("$") f.write(str(n)) f.write("$") f.write( \\\\\n ) f.write( \\\\\n ) for i in range(0,h): # display of the triangle ch="" for j in range(0,i+1): ch+="" + str(t[i][j]) ch=ch+ "\\\\\n" f.write(ch) f.close() # closing of the file 2 Variation of n at r fixed Here, we fix r = 2 and make n vary to study π 2 modulo n. First, let s remind that if n is a prime number, we proved in [2] that : S r (m, k) 0 [n] for all S r (m, k) belonging to the modular angle associated to n, A r (n), which is the whole of the elements of π r ranging in the area between δ r (n) and ρ r (n), except δ r (n) (see Figure 0 below for the case r = 3). As a reminder, ρ r (n) is the line in π r associated with a natural integer n - such that the coordinates (m, k) of each term in this line verify m = n (r 1)(k 1) - and δ r (n) the line which forms an angle of π 4 with the horizontal. The algebraic value of the modular angle is : α = π 4 + arctan(r 1) [2]. 4
ρ r (n) Figure 0 : r=3 A r (n) α δ r (n) This property was shown invertible in [3], provided that n (r 1)! = 1, and generalizable to σ r with a few properties. Let s move on to the simulations. FIGURE 1 π 2 mod 3 FIGURE 2 π 2 mod 4 5
FIGURE 3 π 2 mod 5 FIGURE 4 π 2 mod 6 FIGURE 5 π 2 mod 7 FIGURE 6 π 2 mod 13 6
FIGURE 7 π 2 mod 20 The triangles show different patterns according to whether n is prime (Figures 1, 3) or not (Figures 2, 4). Zeros regions are much denser for n prime. Besides, the associated lines to n contain very few or none zeros if n is not prime, and all zeros (excepted the centre of rotation ) otherwise. This is confirmed in Figures 5, 6, 7. We see clearly in such examples the arithmetical behaviour of π 2 according to whether n is a prime or not : in the affirmative case, patterns are very different from those of the so-called Pascal triangle or the Stirling triangles of first and second kind. Indeed, for r 2, modular properties of π r and σ r are no longer concentrated on the horizontal lines like in the last triangles but make modular rotations of angle arctan(r 1) ([3]). Then, one observes in the figures above new modular properties compared to our previous results in [1] [2] [3] : for n prime only, the geometrical patterns of π 2, or modular angles, seem to be repeated every n lines by shifting one column to the right. This phenomenon is also observable in π r and σ r for r > 2 (see below). 3 Variation of r at n fixed Here, we fix n = 17. 7
FIGURE 8 π 1 mod 17 FIGURE 9 π 2 mod 17 FIGURE 10 π 3 mod 17 FIGURE 11 π 4 mod 17 8
FIGURE 12 π 17 mod 17 FIGURE 13 π 18 mod 17 FIGURE 14 π 19 mod 17 FIGURE 15 π 20 mod 17 The initial case r = 1 is similar to the "Pascal s triangle" modulo n prime (the associated line to n is 9
simply horizontal). Then, Figures 8 to 11 show again the expected rotation of the line associated with n as a function of r and its modular "scanning" of the triangle π r. Let s make here some "zooms" for π r modulo 17 when r varies : FIGURE 16 π 1 mod 17 FIGURE 17 π 2 mod 17 FIGURE 18 π 3 mod 17 FIGURE 19 π 4 mod 17 10
FIGURE 20 π 5 mod 17 FIGURE 21 π 6 mod 17 FIGURE 22 π 9 mod 17 FIGURE 23 π 10 mod 17 As clearly remarkable in those magnifications (Figures 16 to 23), the line (and a fortiori the modular angle) associated to n prime tends to disappear with r increasing, which can be understood in a combinatorial way : as r increases, one can find less and less k-partitions with classes of more than r elements in a set of cardinal n, which implies the limit towards zero with r of each term of π r. This property looks interesting given that the number and size of the required π r terms to characterize their associated prime number n (see theorem above) is decreasing with r. Besides, we observe like a "cyclical" behaviour in terms of r of the congruence properties of π r modulo n : in Figure 12, for n = r = 17, the appearance of π 17 is very singular in that most "zeros" disappear. On the other hand, in Figures 13, 14, 15, one notes that π 18 modulo 17 suddenly finds its zeros (although in a quite different way from π 1 modulo 17) and a modular sweep resumes when r increases, here for r = 18 to 20. Anyway, even isolated zeros seem to adopt this cyclical behaviour in general. The same phenomenon recurs for other n, according to the following Figures 24, 25, 26, 27 : 11
FIGURE 24 π 23 mod 23 FIGURE 25 π 24 mod 23 FIGURE 26 π 29 mod 29 FIGURE 27 π 31 mod 29 12
Finally, sharing the same arithmetical triangle structure as π r [1], triangles σ r modulo n show, not surprisingly, similar properties for n prime. Just a few examples below for n = 11 and r = 1 to r = 4 : FIGURE 28 σ 1 mod 11 FIGURE 29 σ 2 mod 11 FIGURE 30 σ 3 mod 11 FIGURE 31 σ 4 mod 11 13
4 Conclusion Thanks to those simulations, new phenomena in associated Stirling numbers have been observed, in particular cyclical behaviours of π r and σ r modulo n for r n and repeated modular patterns (or angles) along with n. Once proved, such results should be transferred, like we did in [3], to the associated Stirling numbers in the classical staircase form as we find them in the whole literature, i.e. s r (n, k) and d r (n, k), n, k N. Besides, there remain some exciting open questions inspired by this work. For example : Could we find new arithmetical structures with more varied modular angles than defined above, that is to say for any rational, non-positive or even real values of r? To which extent would it be possible to describe the modular properties of π r and σ r (or other similar triangles) in terms of "geometrical" properties only : rotations, translations, asymptotic limits? Are there other arithmetical triangles than π r and σ r or any general structure which have the same kind of modular properties distinct from their geometrical transformations? This would be the object of future papers. Références [1] Ben Letaïef, K., "All Associated Stirling Numbers are Arithmetical Triangles", Notes on Number Theory and Discrete Mathematics, 23, 3, accepted, to be published in September 2017 : https ://hal.archives-ouvertes.fr/hal-01557244 [2] Ben Letaïef, K., "Two Types of Rotations in Associated Stirling Numbers", submitted to Notes on Number Theory and Discrete Mathematics in July 2017 : https ://hal.archives-ouvertes.fr/hal- 01562677 [3] Ben Letaïef, K., "A Disturbing Combination of Geometrical and Modular Rotations in the World of Arithmetic", submitted to Notes on Number Theory and Discrete Mathematics in July 2017 : https ://hal.archives-ouvertes.fr/hal-01568068 [4] Comtet, L., Analyse combinatoire, PUF, 1970. 14