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A260355
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Table T(n,k) read by antidiagonals. T(n,k) is the minimum value of Sum_{i=1..n} Product_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.
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6
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1, 1, 3, 1, 4, 6, 1, 6, 10, 10, 1, 8, 18, 20, 15, 1, 12, 33, 44, 35, 21, 1, 16, 60, 96, 89, 56, 28, 1, 24, 108, 214, 231, 162, 84, 36, 1, 32, 198, 472, 600, 484, 271, 120, 45, 1, 48, 360, 1043, 1564, 1443, 915, 428, 165, 55, 1, 64, 648, 2304, 4074, 4320, 3089, 1608, 642, 220, 66, 1, 96, 1188, 5136, 10618
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OFFSET
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1,3
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COMMENTS
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T(1,k) = 1. T(2,k) = A029744(k+2). T(n,1) = n(n+1)/2 (= A000217(n)). See arXiv link for sets of permutations that achieve the value of T(n,k).
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LINKS
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FORMULA
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T(n,k) >= n*(n!)^(k/n).
If n divides k, then T(n,k) = n*(n!)^(k/n).
T(n,2) = n*(n+1)*(n+2)/6 = A000292(n).
(End)
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EXAMPLE
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(Partially filled in) table starts (with n rows and k columns):
k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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n 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2| 3 4 6 8 12 16 24 32 48 64 96 128 192 256 384
3| 6 10 18 33 60 108 198 360 648 1188 2145 3888 7083 12844 23328
4| 10 20 44 96 214 472 1043 2304 5136 11328 24993 55296 122624 271040 599832
5| 15 35 89 231 600 1564 4074 10618
6| 21 56 162 484 1443 4320
7| 28 84 271 915 3089
8| 36 120 428 1608
9| 45 165 642 2664
10| 55 220 930 4208
11| 66 286 1304
12| 78 364 1781
13| 91 455 2377
14| 105 560 3111
15| 120 680 4002
(Partially filled in) table of how many nonequivalent sets of permutations achieves the value of T(n,k)
k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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n 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3| 1 1 1 1 1 2 1 2 2 2 1 3 1 1 3
4| 1 1 2 4 11 10 10 81 791 533 24 1461 3634 192 2404
5| 1 1 3 12 16 188 211 2685
6| 1 1 10 110 16
7| 1 1 6
8| 1 1 16
9| 1 1 4
10| 1 1 12
11| 1 1
12| 1 1
13| 1 1
14| 1 1
15| 1 1
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PROG
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(Python)
from itertools import permutations, combinations_with_replacement
def A260355(n, k): # compute T(n, k)
if k == 1:
return n*(n+1)//2
ntuple, count = tuple(range(1, n+1)), n**(k+1)
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()
v = 0
for i in range(n):
v += (n-i)*t[i]
if v < count:
count = v
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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