A070735 -id:A070735 - cp's OEIS frontend

A070893 Let r, s, t be three permutations of the set {1,2,3,..,n}; a(n) = value of Sum_{i=1..n} r(i)s(i)t(i), with r={1,2,3,..,n}; s={n,n-1,..,1} and t={n,n-2,n-4,...,1,...,n-3,n-1}.

1, 6, 19, 46, 94, 172, 290, 460, 695, 1010, 1421, 1946, 2604, 3416, 4404, 5592, 7005, 8670, 10615, 12870, 15466, 18436, 21814, 25636, 29939, 34762, 40145, 46130, 52760, 60080, 68136, 76976, 86649, 97206, 108699, 121182, 134710, 149340

Offset: 1

Wouter Meeussen, May 22 2002

See A070735 for the minimal values for these products. This sequence is an upper bound. The third permutation 't'= ceiling(abs(range(n-1/2,-n,-2))) is such that it associates its smallest factor with the largest factor of the product 'r'*'s'.

We observe that is the transform of A002717 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0,u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of v is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010

			{1,2,3,4,5,6,7}*{7,6,5,4,3,2,1}*{7,5,3,1,2,4,6} gives {49,60,45,16,30,48,42}, with sum 290, so a(7)=290.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A070735, A082289. a(n)=A082290(2n-2).
Cf. A000034, A032766, A006578, A002717. - Richard Choulet, Jan 28 2010
Cf. A002717 (first differences). - Bruno Berselli, Aug 26 2011
Column k=3 of A166278. - Alois P. Heinz, Nov 02 2012

[(1/96)*(2*n*(n+2)*(3*n^2+10*n+4)+3*(-1)^n-3): n in [1..40]]; // Vincenzo Librandi, Aug 26 2011

Table[Plus@@(Range[n]*Range[n, 1, -1]*Ceiling[Abs[Range[n-1/2, -n, -2]]]), {n, 49}];
(* or *)
CoefficientList[Series[ -(1+2x)/(-1+x)^5/(1+x), {x, 0, 48}], x]//Flatten

a(n)=sum(i=1,n,i*(n+1-i)*ceil(abs(n+3/2-2*i)))

a(n)=polcoeff(if(n<0,x^4*(2+x)/((1+x)*(1-x)^5),x*(1+2*x)/((1+x)*(1-x)^5))+x*O(x^abs(n)),abs(n))

G.f.: x*(1+2*x)/((1+x)*(1-x)^5). - Michael Somos, Apr 07 2003
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) + 3. If sequence is also defined for n <= 0 by this equation, then a(n)=0 for -3 <= n <= 0 and a(n)=A082289(-n) for n <= -4. - Michael Somos, Apr 07 2003
a(n) = (1/96)*(2*n*(n+2)*(3*n^2+10*n+4)+3*(-1)^n-3). a(n) - a(n-2) = A002411(n). - Bruno Berselli, Aug 26 2011

A070736 Let r, s, t, u be four permutations of the set { 1, 2, 3, ..., n }; a(n) = minimal value of Sum_{i=1..n} r(i)s(i)t(i)*u(i).

Original entry on oeis.org

1, 8, 33, 96, 231, 484, 915, 1608, 2664, 4208, 6392, 9392, 13418, 18706, 25540, 34224, 45108, 58588, 75101, 95120, 119179, 147856, 181786, 221648, 268195, 322220, 384588, 456232, 538138, 631362, 737052, 856396, 990684, 1141254, 1309568, 1497104, 1705508, 1936416, 2191700, 2473248, 2783030

Offset: 1

Michael Reid (mreid(AT)math.umass.edu), May 15 2002

			Examples from _David A. Corneth_, Apr 09 2019:
a(1) = 1 via [1] [1] [1] [1];
a(2) = 8 via [1, 2] [1, 2] [2, 1] [2, 1];
a(3) = 33 via [1, 2, 3] [1, 3, 2] [3, 1, 2] [3, 2, 1];
a(4) = 96 via [1, 2, 3, 4] [2, 1, 4, 3] [3, 4, 1, 2] [4, 3, 2, 1];
a(5) = 231 via [1, 2, 3, 4, 5] [2, 3, 1, 4, 5] [4, 2, 5, 3, 1] [5, 4, 3, 1, 2];
a(6) = 484 via [1, 2, 3, 4, 5, 6] [2, 3, 5, 1, 4, 6] [6, 3, 2, 5, 4, 1] [6, 5, 3, 4, 1, 2].

Martin Fuller, Table of n, a(n) for n = 1..100

Cf. A000292 (for two permutations), A070735 (for three).

Table[Min[Map[Total,Map[#[[1]]*#[[2]]*#[[3]]*#[[4]] &,Subsets[If[n > 3, Permutations[Range[n]],Flatten[Table[Permutations[Range[n]], 4], 1]], {4}]]]] , {n, 1, 5}] (* Robert Price, Apr 09 2019 *)

# See Martin Fuller link in A070735, Aug 06 2023

a(11) onwards from Martin Fuller, Aug 06 2023

A260355 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}.

Original entry on oeis.org

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

Offset: 1

Chai Wah Wu, Jul 29 2015

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).

			(Partially filled in) table starts (with n rows and k columns):
(Third column is A070735, fourth column is A070736)
   k    1   2     3     4     5     6     7     8     9    10    11    12     13     14     15
  --------------------------------------------------------------------------------------------
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
  --------------------------------------------------------------------------------------------
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

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. A001563, A029744, A000217, A000292 (T(n,2)), A070735 (T(n,3)), A070736 (T(n,4)).

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

From Chai Wah Wu, Feb 24 2020: (Start)
T(n,k) >= n*(n!)^(k/n).
If n divides k, then T(n,k) = n*(n!)^(k/n).
T(n,n) = (n+1)! - n! = A001563(n).
T(n,2) = n*(n+1)*(n+2)/6 = A000292(n).
(End)

A072368 Minimal total volume of n bricks with integer sides, all sides being different. Lowest value of sum of products of triples pqr chosen from [1,3n].

Original entry on oeis.org

6, 54, 214, 594, 1334, 2614, 4645, 7676, 11992, 17912, 25791, 36021, 49028, 65269, 85247, 109493, 138575, 173094, 213694, 261048, 315863, 378888, 450907, 532730, 625213, 729244, 845748, 975679, 1120035, 1279848, 1456176, 1650123, 1862831, 2095469, 2349237

Offset: 1

Wouter Meeussen, Jul 19 2002

For n=19, the smallest integer from each triple does not belong to range [1,19]. Triplicating the sets of triples, shifting each triple to the left, generates permutations as in A070735, but not provably minimal ones.

a(n) >= ceiling(n*(3n!)^(1/n)) with the inequality tight for 1 <= n <= 3. - Chai Wah Wu, Mar 05 2020

			a(7)=4645 because (1*20*21)+(2*18*19)+(3*15*16)+(4*13*14)+(5*8*17)+(6*10*12)+(7*9*11)=4645 is the smallest value attainable.

Martin Fuller, Table of n, a(n) for n = 1..80 (terms 1..50 from Rob Pratt)
Martin Fuller, Illustration of initial terms
Martin Fuller, Python program for this sequence
Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Cf. A070735.

Python
```
See Martin Fuller link
```

Corrected and extended via integer linear programming by Rob Pratt, Jul 28 2023

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}.

Original entry on oeis.org

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.

cp's OEIS Frontend

A070893 Let r, s, t be three permutations of the set {1,2,3,..,n}; a(n) = value of Sum_{i=1..n} r(i)*s(i)*t(i), with r={1,2,3,..,n}; s={n,n-1,..,1} and t={n,n-2,n-4,...,1,...,n-3,n-1}.

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A070736 Let r, s, t, u be four permutations of the set { 1, 2, 3, ..., n }; a(n) = minimal value of Sum_{i=1..n} r(i)*s(i)*t(i)*u(i).

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A260355 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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A072368 Minimal total volume of n bricks with integer sides, all sides being different. Lowest value of sum of products of triples p*q*r chosen from [1,3n].

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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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A070893 Let r, s, t be three permutations of the set {1,2,3,..,n}; a(n) = value of Sum_{i=1..n} r(i)s(i)t(i), with r={1,2,3,..,n}; s={n,n-1,..,1} and t={n,n-2,n-4,...,1,...,n-3,n-1}.

A070736 Let r, s, t, u be four permutations of the set { 1, 2, 3, ..., n }; a(n) = minimal value of Sum_{i=1..n} r(i)s(i)t(i)*u(i).

A072368 Minimal total volume of n bricks with integer sides, all sides being different. Lowest value of sum of products of triples pqr chosen from [1,3n].