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Periodicity of the Last Digits of Some Combinatorial Sequences1

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Periodicity of the Last Digits of Some Combinatorial Sequences1 1 2 Journal of Integer Sequences, Vol. 17 (2014), 3 Article 14.1.1 47 6 23 11 Periodicity of the Last Digits of Some Combinatorial Sequences1 Istv´an Mez˝o Escuela Polit´ecnica Nacional Departamento de Matem´atica Ladr´on de Guevara E11-253 Quito, Ecuador [email protected] Abstract In 1962 O. A. Gross proved that the last digits of the Fubini numbers (or surjective numbers) have a simple periodicity property. We extend this result to a wider class of combinatorial numbers coming from restricted set partitions. 1 The Stirling numbers of the second kind and the Fubini numbers The nth Fubini number or surjection number [18, 22, 31, 34, 35], Fn, counts all the possible n partitions of n elements such that the order of the blocks matters. The k Stirling number of the second kind with parameters n and k enumerates the partitions of n elements into k blocks. Thus, Fn has the following expression: n n F = k! . (1) n k k X=0 The first Fubini numbers are presented in the next table: 1This scientific work was financed by Proyecto Prometeo de la Secretar´ıa Nacional de Ciencia, Tecnolog´ıa e Innovaci´on (Ecuador). 1 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 1 3 13 75 541 4683 47293 545 835 7087261 102247563 We may realize that the last digits form a periodic sequence of length four. This is true up to infinity. That is, for all n ≥ 1 Congruence 1. Fn+4 ≡ Fn (mod 10). A proof was given by Gross [20] which uses backward differences. Later we give a simple combinatorial proof. We note that if the order of the blocks does not matter, we get the Bell numbers [11]: n n B = . n k k X=0 This sequence does not possess this “periodicity property”. Now we introduce other classes of numbers which do share this periodicity. 2 The r-Stirling and r-Fubini numbers r n k n The -Stirling number of the second kind with parameters and is denoted by k r and enumerates the partitions of n elements into k blocks such that all the first r elements are in different blocks. So, for example, {1, 2, 3, 4, 5} can be partitioned into {1, 4, 5} ∪ {2, 3} but {1, 2} ∪ {3, 5} ∪ {4} is forbidden, if r ≥ 2. An introductory paper on r-Stirling numbers is due to Broder [5]. A good source of combinatorial identities on r-Stirling numbers is a book of Charalambides [6]. (Note that in this book these numbers are called noncentral Stirling numbers.) Similarly to (1) we can introduce the r-Fubini numbers as n n + r F = (k + r)! . (2) n,r k + r k r X=0 (It is worthwile to shift the indices, since the upper and lower parameters have to be at least r, so n and k can run from zero in the formula.) We note that the literature of these numbers is rather thin. From the analytical point of view they were discussed by Corcino and his co-authors [12]. The present author with Nyul [28] is preparing a manuscript which studies these numbers in a combinatorial way. The first 2-Fubini and 3-Fubini numbers are F1,2 F2,2 F3,2 F4,2 F5,2 F6,2 F7,2 F8,2 10 62 466 4 142 42 610 498 542 6 541 426 95 160 302 and 2 F1,3 F2,3 F3,3 F4,3 F5,3 F6,3 F7,3 F8,3 42 342 3 210 34 326 413 322 5 544 342 82 077 450 1 330 064 406 We see that the sequence of the last digits is also periodic with period four: Congruence 2. Fn+4,r ≡ Fn,r (mod 10) (n,r ≥ 1). If the order of the blocks does not interest us, we get the r-Bell numbers: n n + r B = . n,r k + r k r X=0 These numbers were discussed combinatorially in a paper of the present author [26] and analytically by Corcino and his co-authors [13, 14], and also by Dil and Kurt [15]. The table of these numbers [26] shows that there is possibly no periodicity in the last digits of the r-Bell numbers. 3 Restricted Stirling numbers and three derived se- quences n Let us go further and introduce another class of Stirling numbers. The k ≤m restricted Stirling number of the second kind [1, 8, 9, 10] gives the number of partitions of n elements into k subsets under the restriction that none of the blocks contain more than m elements. The notation reflects this restrictive property. The sum of restricted Stirling numbers gives the restricted Bell numbers [30]: n n B = . n,≤m k k ≤m X=0 Their tables are as follows: B1,≤2 B2,≤2 B3,≤2 B4,≤2 B5,≤2 B6,≤2 B7,≤2 B8,≤2 B9,≤2 B10,≤2 B11,≤2 1 2 4 10 26 76 232 764 2 620 9 496 35 696 B1,≤3 B2,≤3 B3,≤3 B4,≤3 B5,≤3 B6,≤3 B7,≤3 B8,≤3 B9,≤3 B10,≤3 B11,≤3 1 2 5 14 46 166 652 2 780 12 644 61 136 312 676 B1,≤4 B2,≤4 B3,≤4 B4,≤4 B5,≤4 B6,≤4 B7,≤4 B8,≤4 B9,≤4 B10,≤4 B11,≤4 1 2 5 15 51 196 827 3 795 18 755 99 146 556 711 If we eliminate the first elements B1,≤2 and B1,≤3,B2,≤3,B3,≤3 we can see that these sequences, for m =2, 3, seem to be periodic of length five: Congruence 3. Bn,≤2 ≡ Bn+5,≤2 (mod 10) (n> 1). Bn,≤3 ≡ Bn+5,≤3 (mod 10) (n> 3). 3 The third table shows that such kind of congruence cannot be proven if m = 4. The proof of the above congruence for m = 2, 3 is contained in the 5.3 subsection. We will also explain why this property fails to hold whenever m> 3. What happens, if we take restricted Fubini numbers, as n n F = k! ? (3) n,≤m k k ≤m X=0 The next congruences hold. Congruence 4. Fn,≤1 ≡ 0 (mod10) (n> 4), Fn,≤m ≡ 0 (mod10) (n> 4,m =2, 3, 4), (4) Fn,≤m ≡ 0 (mod2) (n>m,m> 4). (5) The first congruence is trivial, since Fn,≤1 = n!. The others will be proved in subsection 5.4. If we consider restricted Stirling numbers of the first kind, a similar conjecture can be phrased. These numbers with parameter n, k and m count all the permutations on n elements with k cycles such that all cycles contain at most m items. Let us denote these numbers by n k ≤m. Then let n n A = . n,≤m k k ≤m X=0 We may call these as restricted factorials, since if m = n (there is no restriction) we get that An,≤m = n!. Note that the sequence (n!) is clearly periodic in the present sense, because n! ≡ 0 (mod 10) if n> 4. The tables A1,≤3 A2,≤3 A3,≤3 A4,≤3 A5,≤3 A6,≤3 A7,≤3 A8,≤3 A9,≤3 A10,≤3 A11,≤3 1 2 6 18 66 276 1 212 5 916 31 068 171 576 1 014 696 A1,≤4 A2,≤4 A3,≤4 A4,≤4 A5,≤4 A6,≤4 A7,≤4 A8,≤4 A9,≤4 A10,≤4 A11,≤4 1 2 6 24 96 456 2 472 14 736 92 304 632 736 4 661 856 suggest that An,≤m is perhaps periodic of order five. The next result can be phrased as follows: Congruence 5. An,≤m ≡ An+5,≤m (mod 10) (n> 2,m =2, 3, 4). Our argument presented in the subsection 5.5 will show that the restricted factorial numbers all terminate with digit 0 if n > 4 and m > 4, this is the reason why we excluded above the case m> 4. Note that An,≤2 = Bn,≤2 and this number equals to the number of involutions on n elements. (Involution is a permutation π such that π2 = 1, the identity permutation.). 4 4 The associated Stirling numbers At the end we turn to the definition of the associated Stirling numbers. The m-associated n k n Stirling number of the second kind with parameters and , denoted by k ≥m, gives the number of partitions of an n element set into k subsets such that every block contains at least m elements [11, p. 221]. The associated Bell numbers are n n B = . n,≥m k k ≥m X=0 The tables for m =2, 3, 4 (Bn,≥1 = Bn, the nth Bell number) follow. B1,≥2 B2,≥2 B3,≥2 B4,≥2 B5,≥2 B6,≥2 B7,≥2 B8,≥2 B9,≥2 B10,≥2 B11,≥2 0 1 1 4 11 41 162 715 3 425 17 722 98 253 B1,≥3 B2,≥3 B3,≥3 B4,≥3 B5,≥3 B6,≥3 B7,≥3 B8,≥3 B9,≥3 B10,≥3 B11,≥3 0 0 1 1 1 11 36 92 491 2 557 11 353 B1,≥4 B2,≥4 B3,≥4 B4,≥4 B5,≥4 B6,≥4 B7,≥4 B8,≥4 B9,≥4 B10,≥4 B11,≥4 0 0 0 1 1 1 1 36 127 337 793 Here one cannot observe periodicity in the last digits. However, this is not the case, if we take the associated Fubini numbers, where the order of the blocks counts: n n F = k! .
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