A070736 Let r, s, t, u be four permutations of the set { 1, 2, 3, ..., n }; a(n) = minimal value of Sum_{i=1..n} r(i)*s(i)*t(i)*u(i).

1, 8, 33, 96, 231, 484, 915, 1608, 2664, 4208, 6392, 9392, 13418, 18706, 25540, 34224, 45108, 58588, 75101, 95120, 119179, 147856, 181786, 221648, 268195, 322220, 384588, 456232, 538138, 631362, 737052, 856396, 990684, 1141254, 1309568, 1497104, 1705508, 1936416, 2191700, 2473248, 2783030

Offset: 1

Author

Michael Reid (mreid(AT)math.umass.edu), May 15 2002

Keywords

• nice
• nonn
• hard

Examples

			Examples from _David A. Corneth_, Apr 09 2019:
a(1) = 1 via [1] [1] [1] [1];
a(2) = 8 via [1, 2] [1, 2] [2, 1] [2, 1];
a(3) = 33 via [1, 2, 3] [1, 3, 2] [3, 1, 2] [3, 2, 1];
a(4) = 96 via [1, 2, 3, 4] [2, 1, 4, 3] [3, 4, 1, 2] [4, 3, 2, 1];
a(5) = 231 via [1, 2, 3, 4, 5] [2, 3, 1, 4, 5] [4, 2, 5, 3, 1] [5, 4, 3, 1, 2];
a(6) = 484 via [1, 2, 3, 4, 5, 6] [2, 3, 5, 1, 4, 6] [6, 3, 2, 5, 4, 1] [6, 5, 3, 4, 1, 2].
		

Links

• Martin Fuller, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000292 (for two permutations), A070735 (for three).

Programs

Mathematica

Table[Min[Map[Total,Map[#[[1]]*#[[2]]*#[[3]]*#[[4]] &,Subsets[If[n > 3, Permutations[Range[n]],Flatten[Table[Permutations[Range[n]], 4], 1]], {4}]]]] , {n, 1, 5}] (* Robert Price, Apr 09 2019 *)

Python

# See Martin Fuller link in A070735, Aug 06 2023

Extensions

a(11) onwards from Martin Fuller, Aug 06 2023