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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}. 2
1, 2, 3, 6, 25, 6, 24, 343, 110, 10, 120, 6561, 3375, 324, 15, 720, 161051, 144400, 17576, 756, 21, 5040, 4826809, 7962624, 1336336, 64000, 1521, 28, 40320, 170859375, 535387328, 130691232, 7595536, 185193, 2756, 36, 3628800, 6975757441 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A dual sequence to A331889.

   k         1          2         3         4       5        6         7       8       9

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

n  1|        1          3         6        10      15       21        28      36      45

   2|        2         25       110       324     756     1521      2756    4624    7310

   3|        6        343      3375     17576   64000   185193    456533 1000000 2000376

   4|       24       6561    144400   1336336 7595536 31640625 106131204

   5|      120     161051   7962624 130691232

   6|      720    4826809 535387328

   7|     5040  170859375

   8|    40320 6975757441

   9|  3628800

  10| 39916800

LINKS

Table of n, a(n) for n=1..38.

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

FORMULA

T(n,k) <= floor((k*(k*n+1)/2)^n) with equality if k = 2*t+n*u for nonnegative integers t and u.

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

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

T(n,2) = (2*n+1)^n = A085527(n).

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).

PROG

(Python)

from itertools import combinations, permutations

from sympy import factorial

def T(n, k): # T(n, k) for A333420

    if k == 1:

        return int(factorial(n))

    if n == 1:

        return k*(k+1)//2

    if k % 2 == 0 or (k >= n-1 and n % 2 == 1):

        return (k*(k*n+1)//2)**n

    if k >= n-1 and n % 2 == 0 and k % 2 == 1:

        return ((k**2*(k*n+1)**2-1)//4)**(n//2)

    nk = n*k

    nktuple = tuple(range(1, nk+1))

    nkset = set(nktuple)

    count = 0

    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()

            w = 1

            for i in range(n):

                w *= llist[i]+t[i]

            if w > count:

                count = w

    return count

CROSSREFS

Cf. A000142, A000217, A085527, A331889.

Sequence in context: A099000 A032540 A063728 * A296259 A344935 A000341

Adjacent sequences:  A333417 A333418 A333419 * A333421 A333422 A333423

KEYWORD

nonn,more,tabl

AUTHOR

Chai Wah Wu, Mar 23 2020

STATUS

approved

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Last modified August 18 02:34 EDT 2022. Contains 356204 sequences. (Running on oeis4.)