A060963 -id:A060963 - cp's OEIS frontend

A261517 Number of perfect rhythmic tilings of [0,4n-1] with quadruplets.

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 24, 38, 96, 444, 1414, 5134, 19490

Offset: 0

Michel Marcus, Aug 23 2015

A perfect tiling of the line with quadruplets consists of groups of four evenly spaced points, each group having a different common interval such that all points of the line are covered.

			For n=1, there is 1 such tiling: (0,1,2,3).
For n=15, there are 2 such tilings: [0, 16, 32, 48], [1, 3, 5, 7], [2, 13, 24, 35], [4, 22, 40, 58], [6, 21, 36, 51], [8, 14, 20, 26], [9, 10, 11, 12], [15, 29, 43, 57], [17, 25, 33, 41], [18, 30, 42, 54], [19, 23, 27, 31], [28, 37, 46, 55], [34, 39, 44, 49], [38, 45, 52, 59], [47, 50, 53, 56] and its mirror (see Ekhad et al. link).

J. P. Delahaye, La musique mathématique de Tom Johnson, in Mathématiques pour le plaisir, Belin-Pour la Science, Paris, 2010.

J. P. Delahaye, La musique mathématique de Tom Johnson, Pour la Science, 325, Nov 2004, pp.88-93.
Shalosh B. Ekhad, Lara Pudwell and Doron Zeilberger, A Perfect Rhythmic Tiling Of Quadruplets, Nov. 30, 2004; Local copy, pdf file only, no active links
Tom Johnson, Perfect Rhythmic Tilings, Lecture delivered at MaMuX meeting, IRCAM, January 24, 2004; Local copy, pdf file only, no active links
Tom Johnson, Tiling in My music, August, 2008; Local copy, pdf file only, no active links

Cf. A060963, A261516.

a(21)-a(23) from Fausto A. C. Cariboni, Mar 18 2017

a(0)=1 prepended by Seiichi Manyama, Feb 22 2020

A322178 The number of permutations of {1,2,...,n,1,2,...,n} with the property that b(1) >= b(2) >= ... >= b(n) (there are b(k) numbers between the two k's for k=1..n).

Original entry on oeis.org

1, 1, 5, 33, 329, 3825, 57293, 977581, 19619645, 442155529, 11183272973, 312134648549, 9554405887621, 317670072938621, 11411690507968361, 440231352579839965

Offset: 0

Seiichi Manyama, Nov 30 2018

			In case of n = 2.
     |              | b(1),b(2)
-----+--------------+----------
   1 | [1, 1, 2, 2] | [0, 0]
   2 | [1, 2, 1, 2] | [1, 1]
   3 | [1, 2, 2, 1] | [2, 0] *
   4 | [2, 1, 2, 1] | [1, 1]
   5 | [2, 2, 1, 1] | [0, 0]
In case of n = 3.
     |                    | b(1),b(2),b(3)
-----+--------------------+---------------
   1 | [1, 1, 2, 2, 3, 3] | [0, 0, 0]
   2 | [1, 1, 3, 3, 2, 2] | [0, 0, 0]
   3 | [1, 2, 1, 2, 3, 3] | [1, 1, 0]
   4 | [1, 2, 2, 1, 3, 3] | [2, 0, 0]
   5 | [1, 2, 2, 3, 3, 1] | [4, 0, 0]
   6 | [1, 2, 3, 1, 2, 3] | [2, 2, 2]
   7 | [1, 2, 3, 2, 3, 1] | [4, 1, 1]
   8 | [1, 2, 3, 3, 1, 2] | [3, 3, 0]
   9 | [1, 2, 3, 3, 2, 1] | [4, 2, 0] *
  10 | [1, 3, 2, 1, 3, 2] | [2, 2, 2]
  11 | [1, 3, 2, 3, 1, 2] | [3, 2, 1] *
  12 | [1, 3, 2, 3, 2, 1] | [4, 1, 1]
  13 | [1, 3, 3, 1, 2, 2] | [2, 0, 0]
  14 | [1, 3, 3, 2, 1, 2] | [3, 1, 0] *
  15 | [1, 3, 3, 2, 2, 1] | [4, 0, 0]
  16 | [2, 1, 2, 1, 3, 3] | [1, 1, 0]
  17 | [2, 1, 2, 3, 1, 3] | [2, 1, 1]
  18 | [2, 1, 2, 3, 3, 1] | [3, 1, 0] *
  19 | [2, 1, 3, 2, 1, 3] | [2, 2, 2]
  20 | [2, 1, 3, 2, 3, 1] | [3, 2, 1] *
  21 | [2, 1, 3, 3, 2, 1] | [3, 3, 0]
  22 | [2, 2, 1, 1, 3, 3] | [0, 0, 0]
  23 | [2, 2, 1, 3, 3, 1] | [2, 0, 0]
  24 | [2, 2, 3, 3, 1, 1] | [0, 0, 0]
  25 | [2, 3, 1, 2, 3, 1] | [2, 2, 2]
  26 | [3, 1, 2, 3, 1, 2] | [2, 2, 2]
  27 | [3, 1, 3, 2, 1, 2] | [2, 1, 1]
  28 | [3, 2, 1, 3, 2, 1] | [2, 2, 2]
  29 | [3, 3, 1, 1, 2, 2] | [0, 0, 0]
  30 | [3, 3, 1, 2, 1, 2] | [1, 1, 0]
  31 | [3, 3, 1, 2, 2, 1] | [2, 0, 0]
  32 | [3, 3, 2, 1, 2, 1] | [1, 1, 0]
  33 | [3, 3, 2, 2, 1, 1] | [0, 0, 0]
* (Strongly decreasing)

Edward Moody, Java program for calculating entries in this sequence and A060963

Cf. A060963 (Strongly decreasing).

a(9) from Seiichi Manyama, Dec 31 2019
a(10)-a(11) from Giovanni Resta, Jan 15 2020
a(12)-a(15) from Edward Moody, Feb 17 2020

A261516 Number of perfect rhythmic tilings of [0,3n-1] with triples.

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 0, 18, 66, 382, 1104, 4138, 15324, 61644, 325456, 2320948, 17660110, 148271962, 1171109228, 9257051746

Offset: 0

Michel Marcus, Aug 23 2015

A perfect tiling of the line with triples consists of groups of three evenly spaced points, each group having a different common interval such that all points of the line are covered.

			For n=1, there is 1 such tiling: (0,1,2).
For n=5, there are 2 such tilings: (2,3,4), (8,10,12), (5,9,13), (1,6,11), (0,7,14) and its mirror, that have these distinct common differences: 1,2,4,5,7.

J. P. Delahaye, La musique mathématique de Tom Johnson, in Mathématiques pour le plaisir, Belin-Pour la Science, Paris, 2010.

J. P. Delahaye, La musique mathématique de Tom Johnson, Pour la Science, 325, Nov 2004, pp. 88-93.
Tom Johnson, Perfect Rhythmic Tilings, Lecture delivered at MaMuX meeting, IRCAM, January 24, 2004.
Tom Johnson, Tiling in My music, August, 2008.

Cf. A060963, A104429, A261517, A285527.

a(16)-a(17) from Alois P. Heinz, Sep 16 2015
a(18)-a(19) from Fausto A. C. Cariboni, Mar 27 2017
a(0)=1 prepended by Seiichi Manyama, Feb 21 2020

A272363 Number of ways to group the first 2n natural numbers into n pairs (xi,yi) with yi>xi, and such that the 2n numbers xi+yi and xi-yi are all different.

Original entry on oeis.org

1, 1, 0, 2, 12, 64, 220, 1886, 16346, 142420, 1302106, 14467384, 177079358

Offset: 0

Michel Marcus, Apr 27 2016

			For n=3, ((1,5), (2,3), (4,6)) is an instance of such grouping. ((2,3), (1,5), (4,6)) is considered to be the same grouping. The other one is ((1,3), (2,6), (4,5)). So a(3) = 2.

Mok-Kong Shen and Tsen-Pao Shen, Number theory Research Problem 39, Bull. Amer. Math. Soc. 68 (1962), 557.

Cf. A002968, A007631, A060963.

okperm(vp, n) = {for (k=1, n-1, if (vp[k] > vp[k+1], return (0));); for (k=1, n, if (vp[k+n] <= vp[k], return (0));); 1;}
a(n) = {nb = 0; nn = 2*n; for (j=0, nn!-1, vp = numtoperm(nn, j); if (okperm(vp, n), vs = vector(n, k, vp[k]+vp[k+n]); vd = vector(n, k, vp[k]-vp[k+n]); if (#vs + #vd == #Set(concat(vs, vd)), nb++););); nb;}

from sympy.utilities.iterables import multiset_partitions
def A272363(n):
    return 1 if n == 0 else sum(1 for p in multiset_partitions(list(range(1,2*n+1)),n) if max(len(d) for d in p) == 2 and len(set([sum(d) for d in p]))+len(set([abs(d[0]-d[1]) for d in p])) == 2*n) # Chai Wah Wu, Oct 08 2018

a(n) >= A002968(n). - Altug Alkan, Oct 05 2018

a(n) <= A060963(n). - Chai Wah Wu, Oct 08 2018

a(0), a(7)-a(10) from Alois P. Heinz, Oct 05 2018

a(11)-a(12) from Giovanni Resta, Oct 11 2018

A320129 Number of ways to group the first 2*n natural numbers into n pairs (xi, yi) such that the n numbers xi + yi are all different.

Original entry on oeis.org

1, 1, 2, 10, 55, 412, 3736, 40518, 505486, 7145031, 112844566, 1970286922, 37676184205

Offset: 0

Altug Alkan, Oct 06 2018

			For n = 2, a(2) = 2 since {(1,3), (2,4)} and {(1,2), (3,4)} are corresponding sets. {(2,4), (1,3)} and {(3,1), (4,2)} are considered to be the same grouping with {(1,3), (2,4)}.

Altug Alkan, Line plot of t(n)-t(n-1) where t(n) = a(n+1)/a(n) for n <= 11

Cf. A002968, A060963, A272363.

okperm(vp, n) = {for (k=1, n-1, if (vp[k] > vp[k+1], return (0)); ); for (k=1, n, if (vp[k+n] <= vp[k], return (0)); ); 1; }
a(n) = if(n==0, 1, {nb = 0; nn = 2*n; for (j=0, nn!-1, vp = numtoperm(nn, j); if (okperm(vp, n), vs = vector(n, k, vp[k]+vp[k+n]); if (#vs == #Set(concat(vs)), nb++); ); ); nb; } ) \\ after Michel Marcus at A272363

from sympy.utilities.iterables import multiset_partitions
def A320129(n):
    return 1 if n == 0 else sum(1 for p in multiset_partitions(list(range(1,2*n+1)),n) if max(len(d) for d in p) == 2 and len(set(sum(d) for d in p)) == n) # Chai Wah Wu, Oct 08 2018

a(n) >= A272363(n).

a(11)-a(12) from Giovanni Resta, Oct 09 2018

A320168 Number of ways to group the first 2*n positive integers into n pairs (xi, yi) with xi < yi, and such that the n numbers (yi mod xi) are all different.

Original entry on oeis.org

1, 1, 2, 2, 7, 12, 22, 26, 85, 226, 717, 1695, 5071, 14275, 47405, 176747, 638329, 2166516

Offset: 0

Altug Alkan, Oct 07 2018

How does a(n+1)/a(n) behave as n increases?

			a(3) = 2 because {(1,3), (2,5), (4,6)} and {(1,5), (2,3), (4,6)} are corresponding sets.
a(4) = 7 because {(1,6), (2,5), (3,8), (4,7)}, {(1,3), (2,7), (4,6), (5,8)}, {(1,7), (2,3), (4,6), (5,8)}, {(1,3), (2,5), (4,7), (6,8)}, {(1,5), (2,3), (4,7), (6,8)}, {(1,2), (3,7), (4,6), (5,8)}, {(1,2), (3,8), (4,7), (5,6)} are corresponding sets.

David A. Corneth, Illustration for a(4)

Cf. A002968, A060963, A272363, A320129.

a(13)-a(17) from Rémy Sigrist, Oct 07 2018

A320333 a(n) is the number of pairings of the first 2k positive integers into k pairs (xi, yi) such that the k numbers (xi + yi) mod k are all different where k = 2n + 1.

Original entry on oeis.org

1, 9, 145, 4641, 231417, 16770369, 1671395713

Offset: 0

Altug Alkan, Oct 11 2018

It can be seen as interesting that a(t) = A178185(t+1) for 0 <= t <= 3.

			a(1) = 9 because {(1,4), (2,5), (3,6)}, {(1,5), (2,6), (3,4)}, {(1,6), (2,4), (3,5)}, {(1,3), (2,6), (4,5)}, {(1,5), (2,3), (4,6)}, {(1,6), (2,3), (4,5)}, {(1,3), (2,4), (5,6)}, {(1,2), (3,5), (4,6)}, {(1,2), (3,4), (5,6)} are corresponding sets.

Cf. A060963, A178185, A320129.

A320334 Number of ways to group the first 2*n positive integers into n pairs (xi, yi) with xi < yi, and such that the n numbers (yi - xi) mod n are all different.

Original entry on oeis.org

1, 1, 0, 3, 12, 25, 0, 475, 5352, 17157, 0, 896535, 15083680, 68879713, 0

Offset: 0

Altug Alkan, Oct 11 2018

A variant of A060963, a(n) <= A060963(n) by definition.

			a(3) = 3 because {(1,4), (2,6), (3,5)}, {(1,5), (2,4), (3,6)}, {(1,6), (2,5), (3,4)} are corresponding sets.

Rémy Sigrist, C++ program for A320334

Cf. A060963, A320168, A320333.

Terms computed by Rémy Sigrist, Oct 11 2018

A107683 Number of perfect Skolem sets.

Original entry on oeis.org

1, 1, 3, 11, 35, 114, 407, 1486, 5414, 19923, 74230, 278462, 1049318, 3972395, 15101658, 57607431, 220391316, 845366406, 3250192681, 12521965697

Offset: 1

Ralf Stephan, Jun 10 2005

nonn

G. Nordh, Perfect Skolem sequences

Cf. A004075, A060963, A099152.

cp's OEIS Frontend

A261517 Number of perfect rhythmic tilings of [0,4n-1] with quadruplets.

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A322178 The number of permutations of {1,2,...,n,1,2,...,n} with the property that b(1) >= b(2) >= ... >= b(n) (there are b(k) numbers between the two k's for k=1..n).

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A261516 Number of perfect rhythmic tilings of [0,3n-1] with triples.

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A272363 Number of ways to group the first 2*n natural numbers into n pairs (xi,yi) with yi>xi, and such that the 2*n numbers xi+yi and xi-yi are all different.

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A320129 Number of ways to group the first 2*n natural numbers into n pairs (xi, yi) such that the n numbers xi + yi are all different.

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PARI

Python

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A320168 Number of ways to group the first 2*n positive integers into n pairs (xi, yi) with xi < yi, and such that the n numbers (yi mod xi) are all different.

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A320333 a(n) is the number of pairings of the first 2*k positive integers into k pairs (xi, yi) such that the k numbers (xi + yi) mod k are all different where k = 2*n + 1.

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A320334 Number of ways to group the first 2*n positive integers into n pairs (xi, yi) with xi < yi, and such that the n numbers (yi - xi) mod n are all different.

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Extensions

A107683 Number of perfect Skolem sets.

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A272363 Number of ways to group the first 2n natural numbers into n pairs (xi,yi) with yi>xi, and such that the 2n numbers xi+yi and xi-yi are all different.

A320333 a(n) is the number of pairings of the first 2k positive integers into k pairs (xi, yi) such that the k numbers (xi + yi) mod k are all different where k = 2n + 1.