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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}. 3
1, 3, 2, 6, 10, 6, 10, 28, 54, 24, 15, 60, 214, 402, 120, 21, 110, 594, 2348, 3810, 720, 28, 182, 1334, 8556, 32808, 43776, 5040, 36, 280, 2614 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

   k    1   2     3     4     5     6     7     8      9      10       11        12

  ---------------------------------------------------------------------------------

n  1|   1   2     6    24   120   720  5040 40320 362880 3628800 39916800 479001600

   2|   3  10    54   402  3810 43776

   3|   6  28   214  2348 32808

   4|  10  60   594  8556

   5|  15 110  1334

   6|  21 182  2614

   7|  28 280

   8|  36 408

   9|  45 570

  10|  55 770

LINKS

Table of n, a(n) for n=1..31.

C. W. Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

FORMULA

T(n,k) >= ceiling(n*((kn)!)^(1/n)).

T(n,1) = n*(n+1)/2 = A000217(n).

T(1,k) = k! = A000142(k).

T(n,3) = A072368(n).

T(n,2) = n*(n+1)*(2*n+1)/3 = A006331(n).

PROG

(Python)

from itertools import combinations, permutations

from sympy import factorial

def T(n, k): # T(n, k) for A331889

    if k == 1:

        return n*(n+1)//2

    if n == 1:

        return int(factorial(k))

    if k == 2:

        return n*(n+1)*(2*n+1)//3

    nk = n*k

    nktuple = tuple(range(1, nk+1))

    nkset = set(nktuple)

    count = int(factorial(nk))

    for firsttuple in combinations(nktuple, n):

        nexttupleset = nkset-set(firsttuple)

        for s in permutations(sorted(nexttupleset), nk-2*n):

            llist = sorted(nexttupleset-set(s), reverse=True)

            t = list(firsttuple)

            for i in range(0, k-2):

                itn = i*n

                for j in range(n):

                        t[j] *= s[itn+j]

            t.sort()

            v = 0

            for i in range(n):

                v += llist[i]*t[i]

            if v < count:

                count = v

    return count

CROSSREFS

Cf. A000142, A000217, A006331, A072368.

Sequence in context: A072765 A210756 A210748 * A109876 A108284 A095011

Adjacent sequences:  A331886 A331887 A331888 * A331890 A331891 A331892

KEYWORD

nonn,more,tabl

AUTHOR

Chai Wah Wu, Mar 20 2020

STATUS

approved

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)