A260355 -id:A260355 - cp's OEIS frontend

A260356 a(n) is the minimal value of Sum_{i=1..n} Product_{j=1..5} r_j(i), where each r_j is a permutation of {1,2,...,n}.

1, 12, 60, 214, 600, 1443, 3089

Offset: 1

Chai Wah Wu, Jul 28 2015

See Wu link for sets of permutations that achieve the value of a(n).

Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized. arXiv:1508.02934 [math.CO], 2015.

Cf. A000292 (two permutations), A070735 (three permutations), A070736 (four permutations), A260357 (six permutations), A260358 (seven permutations), A260359 (eight permutations), A260355.

A260357 a(n) is the minimal value of Sum_{i=1..n} Product_{j=1..6} r_j(i), where each r_j is a permutation of {1,2,...,n}.

Original entry on oeis.org

1, 16, 108, 472, 1564, 4320

Offset: 1

Chai Wah Wu, Jul 28 2015

See arXiv link for sets of permutations that achieve the value of a(n).

Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized. arXiv:1508.02934 [math.CO], 2015.

Cf. A000292 (two permutations), A070735 (three permutations), A070736 (four permutations), A260356 (five permutations), A260358 (seven permutations), A260359 (eight permutations), A260355.

A260358 a(n) is the minimal value of Sum_{i=1..n} Product_{j=1..7} r_j(i), where each r_j is a permutation of {1,2,...,n}.

Original entry on oeis.org

1, 24, 198, 1043, 4074

Offset: 1

Chai Wah Wu, Jul 28 2015

See arXiv link for sets of permutations that achieve the value of a(n).

Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized. arXiv:1508.02934 [math.CO], 2015.

Cf. A000292 (two permutations), A070735 (three permutations), A070736 (four permutations), A260356 (five permutations), A260357 (six permutations), A260359 (eight permutations), A260355.

A260359 a(n) is the minimal value of Sum_{i=1..n} Product_{j=1..8} r_j(i), where each r_j is a permutation of {1,2,...,n}.

Original entry on oeis.org

1, 32, 360, 2304, 10618

Offset: 1

Chai Wah Wu, Jul 28 2015

See arXiv link for sets of permutations that achieve the value of a(n).

Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized. arXiv:1508.02934 [math.CO], 2015.

Cf. A000292 (two permutations), A070735 (three permutations), A070736 (four permutations), A260356 (five permutations), A260357 (six permutations), A260358 (seven permutations), A260355.

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}.

Original entry on oeis.org

1, 2, 2, 6, 9, 3, 24, 64, 20, 4, 120, 625, 216, 36, 5, 720, 7776, 3136, 512, 56, 6, 5040, 117649, 59049, 10000, 1000, 81, 7, 40320, 2097152, 1331000, 248832, 24336, 1728, 110, 8, 362880, 43046721, 35831808, 7529536, 759375, 50625, 2744, 144, 9, 3628800, 1000000000, 1097199376, 268435456, 28652616, 1889568, 93636, 4096, 182, 10

Offset: 1

Chai Wah Wu, Feb 23 2020

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.

			T(n,k)
   k    1    2     3      4      5      6      7      8      9     10     11     12
  ---------------------------------------------------------------------------------
n  1|   1    2     3      4      5      6      7      8      9     10     11     12
   2|   2    9    20     36     56     81    110    144    182    225    272    324
   3|   6   64   216    512   1000   1728   2744   4096   5832   8000  10648  13824
   4|  24  625  3136  10000  24336  50625  93636 160000 256036 390625 571536 810000

Chai Wah Wu, Table of n, a(n) for n = 1..70
Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized, arXiv:1508.02934 [math.CO], 2015-2020.
Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Cf. A000027, A000142, A000169, A001504, A016743, A016766, A061711, A061718, A260355.

from itertools import permutations, combinations_with_replacement
def A331988(n,k): # compute T(n,k)
    if k == 1:
        count = 1
        for i in range(1,n):
            count *= i+1
        return count
    ntuple, count = tuple(range(1,n+1)), 0
    for s in combinations_with_replacement(permutations(ntuple,n),k-2):
        t = list(ntuple)
        for d in s:
            for i in range(n):
                t[i] += d[i]
        t.sort()
        w = 1
        for i in range(n):
            w *= (n-i)+t[i]
        if w > count:
            count = w
    return count

T(n,n) = (n*(n+1)/2)^n = A061718(n).
T(n,k) <= (k(n+1)/2)^n.
T(1,k) = k = A000027(k).
T(n,1) = n! = A000142(n).
T(2,2m) = 9m^2 = A016766(m).
T(2,2m+1) = (3m+1)*(3m+2) = A001504(m).
T(n,2) = (n+1)^n = A000169(n+1).
T(3,k) = 8k^3 = A016743(k) for k > 1.
If n divides k then T(n,k) = (k*(n+1)/2)^n.
If k is even then T(n,k) = (k*(n+1)/2)^n.
If n is odd and k >= n-1 then T(n,k) = (k*(n+1)/2)^n.
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).

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A260356 a(n) is the minimal value of Sum_{i=1..n} Product_{j=1..5} r_j(i), where each r_j is a permutation of {1,2,...,n}.

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A260357 a(n) is the minimal value of Sum_{i=1..n} Product_{j=1..6} r_j(i), where each r_j is a permutation of {1,2,...,n}.

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A260358 a(n) is the minimal value of Sum_{i=1..n} Product_{j=1..7} r_j(i), where each r_j is a permutation of {1,2,...,n}.

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A260359 a(n) is the minimal value of Sum_{i=1..n} Product_{j=1..8} r_j(i), where each r_j is a permutation of {1,2,...,n}.

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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}.

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