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June 10

Combinatorics/set partition(s) relation to the divisibility of natural numbers

Is there a relation between the divisibility of numbers and combinatorics/set partitions?(Thanks.)--82.137.13.98 (talk) 10:52, 10 June 2017 (UTC)[reply]

See Ramanujan's congruences for number theoretical partitions. For the number of set partitions, aka Bell numbers, OEIS: A000110 lists a few properties. For example, if p is prime then a(p) = 2 (mod p), and more generally a(p^m) ≡ m+1 (mod p). Also a(n) ≠ 0 or 6 (mod 8). --RDBury (talk) 14:40, 10 June 2017 (UTC)[reply]

There is an interesting aspect to be underlined: that of divisibility of expressions involving combinations C(n,k) like the expression (C(2p,p) - 2) when n=2p, k=p, p a prime--82.79.115.105 (talk) 19:56, 16 June 2017 (UTC) number divisible to 2 in this case.--82.79.115.105 (talk) 19:46, 16 June 2017 (UTC)[reply]

From the point of view of mathematics education it is desirable that a natural connection between set theory and number theory be made when the operation of division (especially when the remainder is non-zero) is taught but also when the addition, subtraction and multiplication operations are introduced in elementary mathematics with (semi)concrete models based on set of objects and their cardinalities.--82.79.115.105 (talk) 19:56, 16 June 2017 (UTC)[reply]