This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A333420 #20 Mar 31 2020 10:40:40
%S A333420 1,2,3,6,25,6,24,343,110,10,120,6561,3375,324,15,720,161051,144400,
%T A333420 17576,756,21,5040,4826809,7962624,1336336,64000,1521,28,40320,
%U A333420 170859375,535387328,130691232,7595536,185193,2756,36,3628800,6975757441
%N A333420 Table T(n,k) read by upward antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r[(i-1)*k+j] among all permutations r of {1..kn}.
%C A333420 A dual sequence to A331889.
%C A333420 k 1 2 3 4 5 6 7 8 9
%C A333420 --------------------------------------------------------------------------------------
%C A333420 n 1| 1 3 6 10 15 21 28 36 45
%C A333420 2| 2 25 110 324 756 1521 2756 4624 7310
%C A333420 3| 6 343 3375 17576 64000 185193 456533 1000000 2000376
%C A333420 4| 24 6561 144400 1336336 7595536 31640625 106131204
%C A333420 5| 120 161051 7962624 130691232
%C A333420 6| 720 4826809 535387328
%C A333420 7| 5040 170859375
%C A333420 8| 40320 6975757441
%C A333420 9| 3628800
%C A333420 10| 39916800
%H A333420 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 A333420 T(n,k) <= floor((k*(k*n+1)/2)^n) with equality if k = 2*t+n*u for nonnegative integers t and u.
%F A333420 T(n,1) = n! = A000142(n).
%F A333420 T(1,k) = k*(k+1)/2 = A000217(k).
%F A333420 T(n,2) = (2*n+1)^n = A085527(n).
%F A333420 If n is even, k is odd and k >= n-1, then T(n,k) = ((k^2*(k*n+1)^2-1)/4)^(n/2).
%o A333420 (Python)
%o A333420 from itertools import combinations, permutations
%o A333420 from sympy import factorial
%o A333420 def T(n,k): # T(n,k) for A333420
%o A333420 if k == 1:
%o A333420 return int(factorial(n))
%o A333420 if n == 1:
%o A333420 return k*(k+1)//2
%o A333420 if k % 2 == 0 or (k >= n-1 and n % 2 == 1):
%o A333420 return (k*(k*n+1)//2)**n
%o A333420 if k >= n-1 and n % 2 == 0 and k % 2 == 1:
%o A333420 return ((k**2*(k*n+1)**2-1)//4)**(n//2)
%o A333420 nk = n*k
%o A333420 nktuple = tuple(range(1,nk+1))
%o A333420 nkset = set(nktuple)
%o A333420 count = 0
%o A333420 for firsttuple in combinations(nktuple,n):
%o A333420 nexttupleset = nkset-set(firsttuple)
%o A333420 for s in permutations(sorted(nexttupleset),nk-2*n):
%o A333420 llist = sorted(nexttupleset-set(s),reverse=True)
%o A333420 t = list(firsttuple)
%o A333420 for i in range(0,k-2):
%o A333420 itn = i*n
%o A333420 for j in range(n):
%o A333420 t[j] += s[itn+j]
%o A333420 t.sort()
%o A333420 w = 1
%o A333420 for i in range(n):
%o A333420 w *= llist[i]+t[i]
%o A333420 if w > count:
%o A333420 count = w
%o A333420 return count
%Y A333420 Cf. A000142, A000217, A085527, A331889.
%K A333420 nonn,more,tabl
%O A333420 1,2
%A A333420 _Chai Wah Wu_, Mar 23 2020