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Array read by rows, showing the ways of splitting the numbers from 1 to 16 into two groups so that the numbers in each group have the same sum (68) and the same sum of squares (748).
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%I #11 Aug 03 2013 06:42:31

%S 1,2,3,4,7,10,12,13,16,1,2,3,4,8,9,11,14,16,1,2,3,5,6,10,11,14,16,1,2,

%T 3,5,7,8,12,14,16,1,2,4,5,6,8,11,15,16,1,2,7,8,11,12,13,14,1,3,6,8,10,

%U 12,13,15,1,3,6,9,10,11,12,16,1,4,5,8,10,11,14,15,1,4,6,7,9,12,14,15,1,4,6,7,10,11,13,16,1,5,6,7,8,11,14,16

%N Array read by rows, showing the ways of splitting the numbers from 1 to 16 into two groups so that the numbers in each group have the same sum (68) and the same sum of squares (748).

%H Colm Mulcahy, <a href="http://www.maa.org/community/maa-columns/past-columns-card-colm/plurality-events-standard-deviations-and-skewed-perspectives">Plurality Events, Standard Deviations and Skewed Perspectives, Dec 2007</a>

%e There are exactly 657 ways of splitting the numbers 1..16 into two groups so that the numbers in each group have the same sum s1=68.

%e There are exactly 57 ways of splitting the numbers 1..16 into two groups so that the numbers in each group have the same sum of squares s2=748.

%e And there is the only one way of splitting the numbers 1..16 into two groups so that the numbers in each group have the same sum of cubes s3=9248: {{1, 4, 6, 7, 10, 11, 13, 16} (=A133483) and {2, 3, 5, 8, 9, 12, 14, 15} (=A133484)}.

%e Amazingly, this last way also appears in previous two cases! This is given in A133483 and A133484.

%e (Also, there is the only one way of splitting the numbers 1..16 into two groups so that the numbers in each group have the same sum of fourth powers s4=121924: {{1,2,3,4,8,9,10,11,12,16},{5,6,7,13,14,15}}. But this splitting does not give equal sum of powers 1..3.)

%e Intersection of first two cases gives 12 ways of splitting the numbers 1..16 into two groups so that the numbers in each group have the same sum s1=68 and the same sum of squares s2=748;

%e Here we list only groups containing 1 (the corresponding 2nd groups are their complements):

%e {1,2,3,4,8,9,11,14,16},

%e {1,2,3,5,6,10,11,14,16},

%e {1,2,3,5,7,8,12,14,16},

%e {1,2,4,5,6,8,11,15,16},

%e {1,2,7,8,11,12,13,14},

%e {1,3,6,8,10,12,13,15},

%e {1,3,6,9,10,11,12,16},

%e {1,4,5,8,10,11,14,15},

%e {1,4,6,7,9,12,14,15},

%e {1,4,6,7,10,11,13,16},

%e {1,5,6,7,8,11,14,16}.

%e This table read by rows gives the present sequence.

%Y Cf. A133483, A133484, A135418.

%K fini,full,nonn,tabf

%O 1,2

%A _Zak Seidov_, Dec 01 2007

%E Error in example lines corrected by _Colm Mulcahy_, Dec 30 2007