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A331988 Table T(n,k) read by antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.

%I A331988 #37 May 26 2020 15:26:47
%S A331988 1,2,2,6,9,3,24,64,20,4,120,625,216,36,5,720,7776,3136,512,56,6,5040,
%T A331988 117649,59049,10000,1000,81,7,40320,2097152,1331000,248832,24336,1728,
%U A331988 110,8,362880,43046721,35831808,7529536,759375,50625,2744,144,9,3628800,1000000000,1097199376,268435456,28652616,1889568,93636,4096,182,10
%N A331988 Table T(n,k) read by antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.
%C A331988 A dual sequence to A260355. See arXiv link for sets of permutations that achieve the value of T(n,k). The minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] is equal to n!*k^n.
%H A331988 Chai Wah Wu, <a href="/A331988/b331988.txt">Table of n, a(n) for n = 1..70</a>
%H A331988 Chai Wah Wu, <a href="http://arxiv.org/abs/1508.02934">Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized</a>, arXiv:1508.02934 [math.CO], 2015-2020.
%H A331988 Chai Wah Wu, <a href="https://arxiv.org/abs/2002.10514">On rearrangement inequalities for multiple sequences</a>, arXiv:2002.10514 [math.CO], 2020.
%F A331988 T(n,n) = (n*(n+1)/2)^n = A061718(n).
%F A331988 T(n,k) <= (k(n+1)/2)^n.
%F A331988 T(1,k) = k = A000027(k).
%F A331988 T(n,1) = n! = A000142(n).
%F A331988 T(2,2m) = 9m^2 = A016766(m).
%F A331988 T(2,2m+1) = (3m+1)*(3m+2) = A001504(m).
%F A331988 T(n,2) = (n+1)^n = A000169(n+1).
%F A331988 T(3,k) = 8k^3 = A016743(k) for k > 1.
%F A331988 If n divides k then T(n,k) = (k*(n+1)/2)^n.
%F A331988 If k is even then T(n,k) = (k*(n+1)/2)^n.
%F A331988 If n is odd and k >= n-1 then T(n,k) = (k*(n+1)/2)^n.
%F A331988 If n is even and k is odd such that k >= n-1, then T(n,k) = ((k^2*(n+1)^2-1)/4)^(n/2).
%e A331988 T(n,k)
%e A331988    k    1    2     3      4      5      6      7      8      9     10     11     12
%e A331988   ---------------------------------------------------------------------------------
%e A331988 n  1|   1    2     3      4      5      6      7      8      9     10     11     12
%e A331988    2|   2    9    20     36     56     81    110    144    182    225    272    324
%e A331988    3|   6   64   216    512   1000   1728   2744   4096   5832   8000  10648  13824
%e A331988    4|  24  625  3136  10000  24336  50625  93636 160000 256036 390625 571536 810000
%o A331988 (Python)
%o A331988 from itertools import permutations, combinations_with_replacement
%o A331988 def A331988(n,k): # compute T(n,k)
%o A331988     if k == 1:
%o A331988         count = 1
%o A331988         for i in range(1,n):
%o A331988             count *= i+1
%o A331988         return count
%o A331988     ntuple, count = tuple(range(1,n+1)), 0
%o A331988     for s in combinations_with_replacement(permutations(ntuple,n),k-2):
%o A331988         t = list(ntuple)
%o A331988         for d in s:
%o A331988             for i in range(n):
%o A331988                 t[i] += d[i]
%o A331988         t.sort()
%o A331988         w = 1
%o A331988         for i in range(n):
%o A331988             w *= (n-i)+t[i]
%o A331988         if w > count:
%o A331988             count = w
%o A331988     return count
%Y A331988 Cf. A000027, A000142, A000169, A001504, A016743, A016766, A061711, A061718, A260355.
%K A331988 nonn,tabl
%O A331988 1,2
%A A331988 _Chai Wah Wu_, Feb 23 2020