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A331889 Table T(n,k) read by upward antidiagonals. T(n,k) is the minimum value of Sum_{i=1..n} Product_{j=1..k} r[(i-1)*k+j] among all permutations r of {1..kn}.

%I A331889 #25 Jul 24 2023 04:53:42
%S A331889 1,3,2,6,10,6,10,28,54,24,15,60,214,402,120,21,110,594,2348,3810,720,
%T A331889 28,182,1334,8556,32808,43776,5040,36,280,2614
%N A331889 Table T(n,k) read by upward antidiagonals. T(n,k) is the minimum value of Sum_{i=1..n} Product_{j=1..k} r[(i-1)*k+j] among all permutations r of {1..kn}.
%C A331889    k    1   2     3     4     5     6     7     8      9      10       11        12
%C A331889   ---------------------------------------------------------------------------------
%C A331889 n  1|   1   2     6    24   120   720  5040 40320 362880 3628800 39916800 479001600
%C A331889    2|   3  10    54   402  3810 43776
%C A331889    3|   6  28   214  2348 32808
%C A331889    4|  10  60   594  8556
%C A331889    5|  15 110  1334
%C A331889    6|  21 182  2614
%C A331889    7|  28 280
%C A331889    8|  36 408
%C A331889    9|  45 570
%C A331889   10|  55 770
%H A331889 Chai Wah Wu, <a href="https://arxiv.org/abs/2002.10514">On rearrangement inequalities for multiple sequences</a>, arXiv:2002.10514 [math.CO], 2020-2022.
%F A331889 T(n,k) >= ceiling(n*((kn)!)^(1/n)).
%F A331889 T(n,1) = n*(n+1)/2 = A000217(n).
%F A331889 T(1,k) = k! = A000142(k).
%F A331889 T(n,3) = A072368(n).
%F A331889 T(n,2) = n*(n+1)*(2*n+1)/3 = A006331(n).
%o A331889 (Python)
%o A331889 from itertools import combinations, permutations
%o A331889 from sympy import factorial
%o A331889 def T(n,k): # T(n,k) for A331889
%o A331889     if k == 1:
%o A331889         return n*(n+1)//2
%o A331889     if n == 1:
%o A331889         return int(factorial(k))
%o A331889     if k == 2:
%o A331889         return n*(n+1)*(2*n+1)//3
%o A331889     nk = n*k
%o A331889     nktuple = tuple(range(1,nk+1))
%o A331889     nkset = set(nktuple)
%o A331889     count = int(factorial(nk))
%o A331889     for firsttuple in combinations(nktuple,n):
%o A331889         nexttupleset = nkset-set(firsttuple)
%o A331889         for s in permutations(sorted(nexttupleset),nk-2*n):
%o A331889             llist = sorted(nexttupleset-set(s),reverse=True)
%o A331889             t = list(firsttuple)
%o A331889             for i in range(0,k-2):
%o A331889                 itn = i*n
%o A331889                 for j in range(n):
%o A331889                         t[j] *= s[itn+j]
%o A331889             t.sort()
%o A331889             v = 0
%o A331889             for i in range(n):
%o A331889                 v += llist[i]*t[i]
%o A331889             if v < count:
%o A331889                 count = v
%o A331889     return count
%Y A331889 Cf. A000142, A000217, A006331, A072368.
%K A331889 nonn,more,tabl
%O A331889 1,2
%A A331889 _Chai Wah Wu_, Mar 20 2020