Sn Jose Stte University SJSU ScholrWorks Fculty Publictions Mthemtics nd Sttistics Februry 2004 Self-similrity nd symmetries of Pscl s tringles nd simplices mod p Richrd P. Kubelk Sn Jose Stte University, richrd.kubelk@sjsu.edu Follow this nd dditionl works t: https://scholrworks.sjsu.edu/mth_pub Prt of the Mthemtics Commons Recommended Cittion Richrd P. Kubelk. "Self-similrity nd symmetries of Pscl s tringles nd simplices mod p" The Fiboncci Qurterly (2004): 70-75. This Article is brought to you for free nd open ccess by the Mthemtics nd Sttistics t SJSU ScholrWorks. It hs been ccepted for inclusion in Fculty Publictions by n uthorized dministrtor of SJSU ScholrWorks. For more informtion, plese contct scholrworks@sjsu.edu.
SELF-SIMILARITY AND SYMMETRIES OF PASCAL S TRIANGLES AND SIMPLICES MOD p Richrd P. Kubelk Deprtment of Mthemtics, Sn Jose Stte University, Sn Jose, CA 95192-0103 e-mil: kubelk@mth.sjsu.edu (Submitted July 2001-Finl Revision Mrch 2002) 1. INTRODUCTION When one sees printout of Pscl s tringle of binomil coefficients computed modulo the prime 2 (Figure 1, where ll zeros hve been replced by blnks), one is immeditely struck by the plesing self-similrity in the picture. Modulo prime p which is different from 2 (Figure 2), the self-similrity is little more complicted, if no less striking. Another striking feture of the mod 2 tringle (Figure 1) is its bundnt symmetry. Specificlly, ech 2 k+1 -rowed tringulr rry of integers k is invrint under reflections cross its xes of symmetry nd, hence, under rottions of 120 or 240, since these re compositions of reflections. Modulo generl 1 Figure 1: Pscl s tringle mod 2. 1 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 2 1 2 1 2 1 Figure 2: Pscl s tringle mod 3. 70
prime p, we no longer hve this symmetry, but it turns out tht the reflected nd rotted versions of the tringles k do still enjoy the sme kind of self-similrity tht k itself does. In this pper we will investigte the reltionship between self-similrity nd the symmetries of Pscl s tringle mod p. In the process, we will see tht the sme reltionship holds when we look t rbitrry tringulr rrys tht re defined by mod n self-similrity procedure. Moreover, we will lso see tht the sme nlysis cn be pplied to higher-dimensionl rrys of multinomil coefficients, with similr results obtining. And, indeed, the generliztion tht tkes us from Pscl s tringle mod p to rbitrry self-similr tringulr rrys will lso succeed in the higher-dimensionl setting. Self-Similrity Procedure: For fixed prime p, let 0 be the tringulr rry consisting of the p rows t the top of the mod p Pscl s tringle. (For exmple, the top three rows in Figure 2, where p = 3.) If we replce ech entry δ in 0 by δ times 0 copy of 0 ll of whose entries hve been multiplied modulo p by δ we get 1, the tringle composed of the first p 2 rows of Pscl s tringle mod p. Continuing, the replcement of ech entry δ in 0 by δ times copy of k gives k+1, which is composed of the first p k+2 rows of the infinite tringle. On the other hnd, hd we replced ech entry δ in k by copy of δ times 0, we would gin hve gotten k+1. Since these two procedures yield the sme result, we will simply refer to the mod p self-similrity procedure in wht follows. Notice tht for p = 2, 3 yields Figure 1 nd for p = 3, 1 yields Figure 2. This mod p self-similrity of Pscl s tringle, which hs been frequently noted see [5], for exmple is feture of binomil coefficients nd is esily verified using the result of Lucs for computing binomil coefficients modulo primes (Lemm 1 below). The bsic reltionship between the symmetries of Pscl s tringle nd mod p selfsimilrity lluded to bove is given by the following Proposition 1: Let our modulus p be prime. If we denote by Θ k the mod p Pscl s tringle k reflected cross one its xes of symmetry, then Θ k cn lso be obtined by pplying the mod p self-similrity procedure to the seed tringle Θ 0 gotten by subjecting 0 to the sme reflection. The proof depends hevily on the following [2, pp. 417-420]. Lemm 1 (Lucs): Fix prime p. If = 0 + 1 p + + k p k nd b = b 0 + b 1 p + + b k p k with 0 i, b i < p, then ( ) b ( i b i ) mod p. (1) Proof of Proposition 1: Let Θ k be obtined by reflecting k cross its xis of symmetry tht contins its lower left vertex. (The cse where the lower right vertex is fixed is similr, while the cse where the pex is fixed is, of course, trivil.) If we denote by (Θ k ) b the entry in the th row nd the b th column of Θ k, then we cn verify tht (Θ k ) b = ( p k+1 ) 1 b p k+1 1 ( (p k+1 ) 1 b) i (p k+1 1 ) i mod p. (2) [Note tht we number ll rows nd ll columns of our tringles strting with 0 nd tht the columns of tringle re tken to be prllel to its left-hnd edge.] 71
The first eqution in (2) is obtined by inspecting the effect of the reflection on k nd the second follows from Lemm 1. (Here x i denotes the coefficient of p i in the p-dic expnsion of x.) But (p k+1 1) i = p 1 for i = 0,..., k, nd so for ny c < p k+1 we obtin (p k+1 1 c) i = (p 1) c i, since there is no borrowing p-diclly in the subtrction of c from p k+1 1. As result, we now hve (Θ k ) b ( ) (p 1) bi (p 1) i (Θ 0 ) i b i mod p, (3) where the lst equlity comes from n inspection of the effect of the reflection on 0. It now remins to show tht the right-hnd quntity in (3) is ctully the (, b) th entry of the tringle Φ k tht results when we pply the mod p self-similrity construction to Θ 0. Perhps the most elegnt wy of doing this would be to point out tht the self-similrity procedure is relly disguised vrint of the Kronecker tensor product of mtrices modulo p. (See [4, p. 8].) However, it is esy enough to prove the result inductively without this fct. We first define â = k p k nd ˆb = b b k p k when 0, b < p k+1. (So â mod p k nd b ˆb mod p k.) We cn then show tht if â ˆb, (Φ k ) b is the (â, ˆb) th entry of the ( k, b k ) th tringulr block of Φ k, nd so it equls (Θ 0 ) k b k (Φ k 1 )âˆb. We now employ the inductive hypothesis nd (3) to conclude tht (Θ k ) b (Φ k ) b mod p. (Note tht we re using the first of the two equivlent mod p self-similrity procedures.) If â < ˆb then we cn show tht (Φ k ) b lies in one of (Φ k ) s lrgest inverted tringulr holes, which we tke to be filled with zeros. On the other hnd, â < ˆb implies tht i < b i for some i < k nd, therefore, tht (Θ 0 ) i b i = 0. This completes the proof of Proposition 1. 2. TRIANGLES DEFINED BY MOD n SELF-SIMILARITY We re so ccustomed to the symmetry nd grce of Pscl s tringle tht we need little convincing to ccept Proposition 1. Wht is perhps surprising t first glnce is the fct tht it holds for ny tringle Ψ k obtined by pplying the mod n self-similrity procedure to n rbitrry seed tringle Ψ 0. Tht is to sy, insted of strting with tringulr rry of numbers like Pscl s tringle mod p tht possesses mod p self-similrity s one of its properties, we cn tke the mod p self-similrity procedure s our point of deprture. Given ny p-rowed tringle Ψ 0, we cn use it s our seed to generte fmily of tringles Ψ k. (If we wnt to insure tht (Ψ k ) b = (Ψ k+1 ) b =..., nd hence tht Ψ k embeds in Ψ m for k < m thereby justifying the term self-similrity we do need to ssume, however, tht (Ψ 0 ) 00 = 1.) For exmple, Figure 3 shows two seed tringles nd the lrger tringles they generte vi the mod 3 self-similrity procedure. Note tht the left-hnd seed hs 1 s the entry t its pex, but for the right-hnd seed this is not the cse. And, indeed, we see tht the right-hnd seed does not embed in the tringle it genertes. Furthermore, the lrge tringle on the right-hnd side of Figure 3 illustrtes the fct tht there is no unique seed tht genertes given tringle: the three-rowed tringle t its top genertes the entire tringle, but the given seed tringle does lso. In fct, we cn crry out our self-similrity procedure with ny modulus n whether prime or not nd ny n-rowed tringulr seed. We must note, however, tht if our modulus n is not prime, the n k+1 -rowed Pscl s tringle will no longer equl k, the tringulr rry generted modulo n by 0. 72
1 2 2 2 1 2 2 1 2 1 2 1 2 1 1 2 2 2 1 2 2 1 2 1 1 2 2 2 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 2 1 2 2 2 1 2 1 2 2 2 1 2 2 2 2 1 2 1 2 1 2 1 2 1 Figure 3: two seeds nd the tringles they generte mod 3 It turns out tht in the context of tringulr rrys generted by the mod n self-similrity procedure pplied to rbitrry seed tringles, the conclusion of Proposition 1 still holds. Nmely, we hve Proposition 2: Fix n, not necessrily prime. Let Ψ k be obtined by pplying the mod n selfsimilrity procedure to the n-rowed seed tringle Ψ 0. If we denote by Υ k the tringle obtined by subjecting Ψ k to one of the symmetries of n equilterl tringle reflection or suitble rottion then Υ k cn lso be constructed by pplying the mod n self-similrity procedure to the seed tringle Υ 0 gotten by subjecting Ψ 0 to the sme symmetry. Figure 3 provides n illustrtion of Proposition 2: rotting the left-hnd seed by +120 gives the right-hnd seed, which, in turn, genertes the lrger tringle on the right; but tht tringle is just the result of rotting the lrger left-hnd tringle by +120. The key fct used in estblishing Proposition 2 is tht the proof of Proposition 1 relly does not rely on specific properties of binomil coefficients beyond the fct tht we cn use the lemm of Lucs (Lemm 1). But, in fct, hving the pproprite Lucs-type lemm is equivlent to being generted from seed tringle by wht we hve clled the self-similrity process. Tht is, we hve the following Proposition 3: Suppose Ψ k is n n k+1 -rowed tringulr rry. Ψ k is obtined by pplying the mod n self-similrity procedure to the n-rowed seed tringle Ψ 0 if nd only if (Ψ k ) b (Ψ 0 ) i b i mod n (4) for ll, b with 0 b < n k+1. (Here = 0 + 1 n+ + k n k nd b = b 0 +b 1 n+ +b k n k with 0 i, b i < n.) Remrk: Note tht in Proposition 3 we do not require tht (Ψ 0 ) 00 = 1, i.e., tht Ψ 0 embed in Ψ k. Proof: By induction on k. The demonstrtion tht self-similrity implies Lucs-type lemm is given by second prt of the proof of Proposition 1. 73
On the other hnd, if we hve Ψ k tht stisfies Lucs-type lemm, (Ψ k ) b T i b i mod n compre it with the Φ k generted by the mod n self-similrity process from T nd see tht the tringles Ψ k nd Φ k gree entry by entry. 3. MULTINOMIAL COEFFICIENTS AND HIGHER DIMENSIONS Let us now turn to higher dimensions. Putz points out in [3] tht the multinomil coefficients, ( ) b 1,...,b m, cn be rryed in n m-dimensionl figure, which he clls Pscl s polytope. I prefer to use the geometriclly more specific term Pscl s simplex. Nomenclture side, s for Pscl s tringle it is true tht coefficient on the -th level of this rry is the sum of the pproprite ncestors on the ( 1)-st level bove. For multinomil coefficients, this is just the fmilir recurrence reltion ( ) ( ) ( ) 1 1 = + +. b 1,..., b m b 1 1, b 2,..., b m b 1,..., b m 1 But Lucs s result (Lemm 1) hs n m-dimensionl nlogue [1], so it turns out tht k, the p k+1 -leveled m-dimensionl Pscl s simplex modulo prime p, cn be obtined by pplying the obvious extension of our mod p self-similrity procedure to the seed simplex 0 consisting of the first p levels of k. Furthermore, the symmetry results we estblished in the twodimensionl cse generlize esily to m dimensions, to wit: Proposition 4: If we denote by Φ k the simplex obtined by subjecting k, the m-dimensionl mod p Pscl s simplex, to one of the symmetries of n equilterl m-simplex reflection or suitble rottion then Φ k cn lso be gotten by pplying the mod p self-similrity procedure to the seed simplex Φ 0 obtined by subjecting 0 to the sme symmetry. Symmetries of n m-simplex correspond exctly to the permuttions of their m+1 vertices. Since the symmetric group on m+1 letters is generted by simple trnspositions, it is enough to prove our result for the symmetries corresponding to these trnspositions, nmely, reflections in hyperplnes tht contin ll but one of the vertices. And, indeed, for ech trnsposition (ij) we cn write down its reflection τ ij for the p k -leveled m-simplex explicitly: ( ) ( ) τ ij = b 1,..., b i,..., b j,..., b m b 1,..., b j,..., b i,..., b m for 0 < i < j m nd, tking the 0-th vertex of the simplex to be its pex, ( ) ( p k+1 ) 1 b i τ 0i = b 1,..., b i 1, b i, b i+1,..., b m b 1,..., b i 1, p k+1 1, b i+1,..., b m for 1 i m. Given these formule, the proof of Proposition 4 prllels tht of Proposition 1. Once gin we cn see tht the symmetry results of Proposition 4 hold in generl for the m-simplex Φ k obtined by pplying the self-similrity procedure to ny seed simplex Φ 0 : 74
Proposition 5: Fix n, not necessrily prime. Let Ψ k be the m-simplex obtined by pplying the mod n self-similrity procedure to the n-leveled seed simplex Ψ 0. If we denote by Φ k the simplex obtined by subjecting Ψ k to one of the symmetries of n equilterl m-simplex reflection or suitble rottion then Φ k cn lso be constructed by pplying the mod n self-similrity procedure to the seed simplex Φ 0 gotten by subjecting Ψ 0 to the sme symmetry. REFERENCES [1] L.E. Dickson. Theorems on the Residues of Multinomil Coefficients With Respect to Prime Modulus. Qurt. J. Pure Appl. Mth. 33 (1901-2): 378-384. [2] E. Lucs. Théorie des Nombres, Guthier-Villrs, Pris (1891). [3] J.F. Putz. The Pscl Polytope: An Extension of Pscl s Tringle to N Dimensions. Coll. Mth. Jour. 17 (1986): 144-155. [4] J.-P. Serre. Liner Representtions of Finite Groups. Springer, New York (1977). [5] M. Sved. Divisibility with Visibility. Mth. Intellig 10.2 (1988): 56-64. AMS Clssifiction Numbers: 11B65, 05A10 75