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A331988
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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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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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T(n,n) = (n*(n+1)/2)^n = A061718(n).
T(n,k) <= (k(n+1)/2)^n.
T(2,2m+1) = (3m+1)*(3m+2) = A001504(m).
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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EXAMPLE
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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
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PROG
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(Python)
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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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