A045723 -id:A045723 - cp's OEIS frontend

A296143 Number of configurations, excluding reflections and color swaps, of n beads each of three colors on a string.

1, 11, 148, 2955, 63231, 1430912, 33259920, 788827215, 18989544145, 462583897776, 11377251858336, 282061000649064, 7039841561638536, 176714389335432960, 4457914983511649088, 112945455380006673039, 2872488224771372668725, 73301643957476400237200, 1876197202671454764901800, 48152601206547990689466930

Offset: 1

Marko Riedel, Dec 05 2017

nonn

Power Group Enumeration applies here.

E. Palmer and F. Harary, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Unique rows of pebbles
Marko Riedel, Maple code for sequences A045723, A296143, A296144, A296145, A296146, including closed form and enumeration

Cf. A045723, A296144, A296145, A296146.

With Z(S_{q,|m}) = [w^q] exp(Sum_{d|m} a_d w^d/d) and parameters n,k we have for nk even, (1/2) ((nk!)/k!/n!^k + (nk/2)! 2^(nk/2) [a_2^(nk/2)] Z(S_{k,|2})(Z_{n,|2}, a_2^n/n!) and for nk odd, (1/2) ((nk!)/k!/n!^k + ((nk-1)/2)! 2^((nk-1)/2) [a_1 a_2^((nk-1)/2)] Z(S_{k,|2})(Z_{n,|2}, a_2^n/n!). This sequence has k=3.

A296144 Number of configurations, excluding reflections and color swaps, of n beads each of four colors on a string.

Original entry on oeis.org

1, 65, 7780, 1315825, 244448316, 48099214856, 9844135755168, 2074189508907945, 446932339677117580, 98028351499011470680, 21813996435165740009568, 4912693780465467348590056, 1117598703447726807428962400, 256444915320263078585645544000, 59283681793041084579875939892480, 13794224341895239072712767055117865

Offset: 1

Marko Riedel, Dec 05 2017

nonn

Power Group Enumeration applies here.

E. Palmer and F. Harary, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Unique rows of pebbles

Cf. A045723, A296143, A296145, A296146.

With Z(S_{q,|m}) = [w^q] exp(Sum_{d|m} a_d w^d/d) and parameters n,k we have for nk even, (1/2) ((nk!)/k!/n!^k + (nk/2)! 2^(nk/2) [a_2^(nk/2)] Z(S_{k,|2})(Z_{n,|2}, a_2^n/n!) and for nk odd, (1/2) ((nk!)/k!/n!^k + ((nk-1)/2)! 2^((nk-1)/2) [a_1 a_2^((nk-1)/2)] Z(S_{k,|2})(Z_{n,|2}, a_2^n/n!). This sequence has k=4.

A296145 Number of configurations, excluding reflections and color swaps, of n beads each of five colors on a string.

Original entry on oeis.org

1, 513, 701260, 1273147785, 2597337494136, 5711975829039480, 13239412829570653440, 31902976888441563215025, 79210992511055955027177700, 201394898991255834414075013488, 522024491776928458970588283023040, 1374924298868439440732405164346591160, 3670434093979203432106449568933449100800, 9911788665178411118992936004423729374579200

Offset: 1

Marko Riedel, Dec 05 2017

nonn

Power Group Enumeration applies here.

E. Palmer and F. Harary, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Unique rows of pebbles

Cf. A045723, A296143, A296144, A296146.

With Z(S_{q,|m}) = [w^q] exp(Sum_{d|m} a_d w^d/d) and parameters n,k we have for nk even, (1/2) ((nk!)/k!/n!^k + (nk/2)! 2^(nk/2) [a_2^(nk/2)] Z(S_{k,|2})(Z_{n,|2}, a_2^n/n!) and for nk odd, (1/2) ((nk!)/k!/n!^k + ((nk-1)/2)! 2^((nk-1)/2) [a_1 a_2^((nk-1)/2)] Z(S_{k,|2})(Z_{n,|2}, a_2^n/n!). This sequence has k=5.

A296146 Number of configurations, excluding reflections and color swaps, of n beads each of six colors on a string.

Original entry on oeis.org

1, 5363, 95304160, 2254635672135, 61689337799825736, 1854290094982330189184, 59529536963190914931717120, 2006426039057377710970239751995, 70206501544183654687465441723567000, 2530662094366411886472214155427418011488, 93449587615256254621892607439280048712775680

Offset: 1

Marko Riedel, Dec 05 2017

nonn

Power Group Enumeration applies here.

E. Palmer and F. Harary, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Unique rows of pebbles

Cf. A045723, A296143, A296144, A296145.

With Z(S_{q,|m}) = [w^q] exp(Sum_{d|m} a_d w^d/d) and parameters n,k we have for nk even, (1/2) ((nk!)/k!/n!^k + (nk/2)! 2^(nk/2) [a_2^(nk/2)] Z(S_{k,|2})(Z_{n,|2}, a_2^n/n!) and for nk odd, (1/2) ((nk!)/k!/n!^k + ((nk-1)/2)! 2^((nk-1)/2) [a_1 a_2^((nk-1)/2)] Z(S_{k,|2})(Z_{n,|2}, a_2^n/n!). This sequence has k=6.

A045610 Number of different energy states of n positive and n negative charges on a string.

Original entry on oeis.org

1, 1, 3, 7, 22, 70, 249, 880, 3238, 12180, 46247, 174458, 672920, 2585414, 10015955

Offset: 0

Wouter Meeussen

			For n=3 the 20 different arrangements of -1,-1,-1,1,1,1 result in 7 energy levels (sum of signed inverse distances):
{0,0,0,1,1,1},{1,1,1,0,0,0}: 13/10
{0,0,1,0,1,1},{1,1,0,1,0,0}: -41/30
{0,0,1,1,0,1},{0,1,0,0,1,1},{1,0,1,1,0,0},{1,1,0,0,1,0}: -56/30
{0,0,1,1,1,0},{0,1,1,1,0,0},{1,0,0,0,1,1},{1,1,0,0,0,1}: -8/10
{0,1,0,1,0,1},{1,0,1,0,1,0}: -37/10
{0,1,0,1,1,0},{0,1,1,0,1,0},{1,0,0,1,0,1},{1,0,1,0,0,1}: -89/30
{0,1,1,0,0,1},{1,0,0,1,1,0}: -71/30
so the multiplicities are 4*2 + 3*4 = 20 = binomial(6,3).

Sean A. Irvine, Java program, GitHub.

Cf. A045723.

f[li_: {(0 | 1) ..}] := Outer[Times, 2 li - 1, 2 li - 1];
Table[Length @ Tally[Total[1/DeleteCases[f[#] DistanceMatrix[Range[2 n]], 0, 2], 2] & /@ Permutations[Join[Table[0, n], Table[1, n]]]], {n, 10}] (* Wouter Meeussen, Mar 15 2021 *)

Corrected and extended by Wouter Meeussen, Mar 15 2021

a(12)-a(15) from Sean A. Irvine, Mar 15 2021

A045631 Number of 2n-bead black-white reversible complementable strings with n black beads and fundamental period 2n.

Original entry on oeis.org

1, 1, 2, 6, 20, 70, 243, 889, 3276, 12276, 46435, 176869, 677022, 2602197, 10033212, 38787495, 150286400, 583434322, 2268848988, 8836447021, 34461894010, 134564992002, 526025788992, 2058359779051, 8061905110548, 31602659997975

Offset: 0

David W. Wilson

nonn

Moebius transform of A045723 (Christian Bower).

cp's OEIS Frontend

A296143 Number of configurations, excluding reflections and color swaps, of n beads each of three colors on a string.

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A296144 Number of configurations, excluding reflections and color swaps, of n beads each of four colors on a string.

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A296145 Number of configurations, excluding reflections and color swaps, of n beads each of five colors on a string.

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A296146 Number of configurations, excluding reflections and color swaps, of n beads each of six colors on a string.

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A045610 Number of different energy states of n positive and n negative charges on a string.

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A045631 Number of 2n-bead black-white reversible complementable strings with n black beads and fundamental period 2n.

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Author

Keywords

Formula