[math-fun] polygonal sets of numbers
A friend of mine is seeking a name for a set (or multiset) of numbers that has the property (*) every number in the set is no larger than the sum of the other numbers in the set. It is easy to see that these numbers (assumed real) are those that arise as the lengths of a (possibly degenerate) polygon. I know that polygonal numbers sometimes refer to triangular, square, pentagonal,..., integers. But we cannot think of a better name for a set or multiset satisfying (*). Hoping to find something in the OEIS: I wrote a program to compute for positive integer n the number of non-empty subsets of {1,2,...,n} satisfying (*) and obtained the following sequence for n from 3 to 14. 1, 4, 13, 35, 85, 194, 425, 904, 1885, 3878, 7904, 16008 This is not in the OEIS. However, the sequence a where a(n) is the number of 3-element subsets of {1,...,n} satisfying (*) is in the OEIS as http://www.research.att.com/projects/OEIS?Anum=A002623 We welcome comments. --Edwin
Date: Tue, 13 Jul 2004 13:03:53 -0400 (EDT) From: Edwin Clark <eclark@math.usf.edu>
A friend of mine is seeking a name for a set (or multiset) of numbers that has the property
(*) every number in the set is no larger than the sum of the other numbers in the set.
Call such sets "weakly egalitarian" - like the US. Not even Bill Gates has most of of the money in the country - as yet.
John McCarthy wrote:
Edwin Clark wrote:
A friend of mine is seeking a name for a set (or multiset) of numbers that has the property
(*) every number in the set is no larger than the sum of the other numbers in the set.
Call such sets "weakly egalitarian" - like the US. Not even Bill Gates has most of of the money in the country - as yet.
Thinking of John McCarthy's suggestion, I just submitted the following sequence to the OEIS 1, 3, 6, 11, 18, 28, 42, 61, 86, 119, 162, 217, 287, 375 The nth term is the number of subsets S of {1,2,...,n} which contain a number that is greater than the sum of the other numbers in S. Actually this is just 2^n-1-b(n) where b(n) is the number of "weakly egalitarian" non-empty subsets of {1,...,n}. But somehow it seems simpler. Perhaps someone can find a nice formula for the nth term. --Edwin
At 06:16 PM 7/13/2004, Edwin Clark wrote:
Thinking of John McCarthy's suggestion, I just submitted the following sequence to the OEIS
1, 3, 6, 11, 18, 28, 42, 61, 86, 119, 162, 217, 287, 375 1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 43, 55, 70, 88 first differences 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18 second differences
The nth term is the number of subsets S of {1,2,...,n} which contain a number that is greater than the sum of the other numbers in S. Actually this is just 2^n-1-b(n) where b(n) is the number of "weakly egalitarian" non-empty subsets of {1,...,n}. But somehow it seems simpler.
Perhaps someone can find a nice formula for the nth term.
The second differences are sequence A000009, partitions into distinct parts. This is easy to see. Let k be the largest element (the "dictator") in S, and let j be the sum of the remaining elements, so 0 <= j < k. For a given k and j, the number of subsets S is just the number of partitions j into distinct parts; call that a(j). Then b(n) = Sum_{1<=k<=n} Sum_{0<=j<k} a(j). -- Fred W. Helenius <fredh@ix.netcom.com>
participants (3)
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Edwin Clark -
Fred W. Helenius -
John McCarthy