A305752 -id:A305752 - cp's OEIS frontend

A056273 Word structures of length n using a 6-ary alphabet.

1, 1, 2, 5, 15, 52, 203, 876, 4111, 20648, 109299, 601492, 3403127, 19628064, 114700315, 676207628, 4010090463, 23874362200, 142508723651, 852124263684, 5101098232519, 30560194493456, 183176170057707, 1098318779272060, 6586964947803695, 39510014478620232, 237013033135668883

Offset: 0

Marks R. Nester

Set partitions of the n-set into at most 6 parts; also restricted growth strings (RGS) with six letters s(1),s(2),...,s(6) where the first occurrence of s(j) precedes the first occurrence of s(k) for all j < k. - Joerg Arndt, Jul 06 2011
Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.
Density of regular language L over {1,2,3,4,5,6}^* (i.e., number of strings of length n in L) described by regular expression with c=6: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*) where Sum stands for union and Product for concatenation. - Nelma Moreira, Oct 10 2004
Word structures of length n using an N-ary alphabet are generated by taking M^n* the vector [(N 1's),0,0,0,...], leftmost column term = a(n+1). In the case of A056273, the vector = [1,1,1,1,1,1,0,0,0,...]. As the vector approaches all 1's, the leftmost column terms approach A000110, the Bell sequence. - Gary W. Adamson, Jun 23 2011
From Gary W. Adamson, Jul 06 2011: (Start)
Construct an infinite array of sequences representing word structures of length n using an N-ary alphabet as follows:
.
  1, 1, 1,  1,  1,   1,   1,    1, ...; N=1, A000012
  1, 2, 4,  8, 16,  32,  64,  128, ...; N=2, A000079
  1, 2, 5, 14, 41, 122, 365, 1094, ...; N=3, A007051
  1, 2, 5, 15, 51, 187, 715, 2795, ...; N=4, A007581
  1, 2, 5, 15, 52, 202, 855, 3845, ...; N=5, A056272
  1, 2, 5, 15, 52, 203, 876, 4111, ...; N=6, A056273
  ...
The sequences tend to A000110. Finite differences of columns reinterpreted as rows generate A008277 as a triangle: (1; 1,1; 1,3,1; 1,7,6,1; ...). (End)

			For a(4) = 15, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD; the 8 chiral patterns are the 4 pairs AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Muniru A Asiru, Table of n, a(n) for n = 0..1275
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Olli-Samuli Lehmus, Optimized Static Allocation of Signal Processing Tasks onto Signal Processing Cores, Master's Thesis, Aalto Univ. (Finland, 2023). See p. 35.
Nelma Moreira and Rogerio Reis, dcc-2004-07.ps
Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Index entries for linear recurrences with constant coefficients, signature (16,-95,260,-324,144).

A row of the array in A278984 and A320955.
Cf. A000351, A000400, A007581, A056272, A008290, A007051, A099263, A099262, A000110, A008277.
Cf. A056325 (unoriented), A320936 (chiral), A305752 (chiral).

List([0..25],n->Sum([0..6],k->Stirling2(n,k))); # Muniru A Asiru, Oct 30 2018

[(&+[StirlingSecond(n, i): i in [0..6]]): n in [0..30]]; // Vincenzo Librandi, Nov 07 2018

egf := (265+264*exp(x)+135*exp(x*2)+40*exp(x*3)+15*exp(x*4)+exp(6*x))/6!:
ser := series(egf,x,30): seq(n!*coeff(ser,x,n),n=0..22); # Peter Luschny, Nov 06 2018

Table[Sum[StirlingS2[n,k],{k,0,6}],{n,0,30}] (* or *) LinearRecurrence[ {16,-95,260,-324,144},{1,1,2,5,15,52},30] (* Harvey P. Dale, Jun 05 2015 *)

Vec((1 - 15*x + 81*x^2 - 192*x^3 + 189*x^4 - 53*x^5)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-6*x)) + O(x^30)) \\ Michel Marcus, Nov 07 2018

a(n) = Sum_{k=0..6} Stirling2(n, k).
For n > 0, a(n) = (1/6!)*(6^n + 15*4^n + 40*3^n + 135*2^n + 264). - Vladeta Jovovic, Aug 17 2003
From Nelma Moreira, Oct 10 2004: (Start)
For n > 0 and c = 6:
a(n) = (c^n)/c! + Sum_{k=0..c-2} ((k^n)/k!*(Sum_{j=2..c-k}(((-1)^j)/j!))).
a(n) = Sum_{k=1..c} (g(k, c)*k^n) where g(1, 1) = 1; g(1, c) = g(1, c-1) + ((-1)^(c-1))/(c-1)! if c>1. For 2 <= k <= c: g(k, c) = g(k-1, c-1)/k if c>1.  (End)
G.f.: (1 - 15*x + 81*x^2 - 192*x^3 + 189*x^4 - 53*x^5)/((1-x)*(1-2x)*(1-3x)*(1-4x)*(1-6x)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 [corrected by R. J. Mathar, Sep 16 2009] [Adapted to offset 0 by Robert A. Russell, Nov 06 2018]
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=6. - Robert A. Russell, Apr 25 2018
E.g.f.: (265 + 264*exp(x) + 135*exp(x*2) + 40*exp(x*3) + 15*exp(x*4) + exp(6*x))/6!. - Peter Luschny, Nov 06 2018

a(0)=1 prepended by Robert A. Russell, Nov 06 2018

A056325 Number of reversible string structures with n beads using a maximum of six different colors.

Original entry on oeis.org

1, 1, 2, 4, 11, 32, 117, 467, 2135, 10480, 55091, 301633, 1704115, 9819216, 57365191, 338134521, 2005134639, 11937364184, 71254895955, 426063226937, 2550552314219, 15280103807200, 91588104196415, 549159428968825

Offset: 0

Marks R. Nester

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure. Thus aabc, cbaa and bbac are all considered to be identical.
Number of set partitions of an unoriented row of n elements with six or fewer nonempty subsets. - Robert A. Russell, Oct 28 2018
There are nonrecursive formulas, generating functions, and computer programs for A056273 and A305752, which can be used in conjunction with the first formula. - Robert A. Russell, Oct 28 2018
From Allan Bickle, Jun 23 2022: (Start)
a(n) is the number of (unlabeled) 6-paths with n+8 vertices. (A 6-path with order n at least 8 can be constructed from a 6-clique by iteratively adding a new 6-leaf (vertex of degree 6) adjacent to an existing 6-clique containing an existing 6-leaf.)
Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)

			For a(4)=11, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, ABBC, and ABCD.  The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Colin Barker, Table of n, a(n) for n = 0..1000
Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
L. Markenzon, O. Vernet, and P. R. da Costa Pereira, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
Index entries for linear recurrences with constant coefficients, signature (16,-84,84,685,-2140,180,7200,-8244,-4176,11664,-5184).

Cf. A056308.
Column 6 of A320750.
Cf. A056273 (oriented), A320936 (chiral), A305752 (achiral).
The numbers of unlabeled k-paths for k = 2..7 are given in A005418, A001998, A056323, A056324, A056325, and A345207, respectively.
The sequences above converge to A103293(n+1).

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=6; Table[Sum[StirlingS2[n,j]+Ach[n,j],{j,0,k}]/2,{n,0,40}] (* Robert A. Russell, Oct 28 2018 *)
LinearRecurrence[{16, -84, 84, 685, -2140, 180, 7200, -8244, -4176, 11664, -5184}, {1, 1, 2, 4, 11, 32, 117, 467, 2135, 10480, 55091, 301633}, 40] (* Robert A. Russell, Oct 28 2018 *)

Vec((1 - 15*x + 70*x^2 - 28*x^3 - 654*x^4 + 1479*x^5 + 783*x^6 - 5481*x^7 + 3512*x^8 + 4640*x^9 - 5922*x^10 + 1530*x^11) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)) + O(x^30)) \\ Colin Barker, Apr 15 2020

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
From Robert A. Russell, Oct 28 2018: (Start)
a(n) = (A056273(n) + A305752(n)) / 2.
a(n) = A056273(n) - A320936(n) = A320936(n) + A305752(n).
a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=6 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n) = A000007(n) + A057427(n) + A056326(n) + A056327(n) + A056328(n) + A056329(n) + A056330(n).
(End)
From Colin Barker, Mar 24 2020: (Start)
G.f.: (1 - 15*x + 70*x^2 - 28*x^3 - 654*x^4 + 1479*x^5 + 783*x^6 - 5481*x^7 + 3512*x^8 + 4640*x^9 - 5922*x^10 + 1530*x^11) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)).
a(n) = 16*a(n-1) - 84*a(n-2) + 84*a(n-3) + 685*a(n-4) - 2140*a(n-5) + 180*a(n-6) + 7200*a(n-7) - 8244*a(n-8) - 4176*a(n-9) + 11664*a(n-10) - 5184*a(n-11) for n>11.
(End)
From Allan Bickle, Jun 23 2022: (Start)
a(n) = (1/1440)*6^n + (1/96)*4^n + (1/36)*3^n + (3/32)*2^n + (19/360)*6^(n/2) + (1/9)*3^(n/2) + (1/8)*2^(n/2) + 17/60 for n > 0 even;
a(n) = (1/1440)*6^n + (1/96)*4^n + (1/36)*3^n + (3/32)*2^n + (13/720)*6^((n+1)/2) + (1/18)*3^((n+1)/2) + (1/16)*2^((n+1)/2) + 17/60 for n > 0 odd. (End)

Another term from Robert A. Russell, Oct 29 2018

a(0)=1 prepended by Robert A. Russell, Nov 09 2018

A305749 T(n,k) is the number of achiral color patterns (set partitions) in a row or loop of length n with k or fewer colors (sets).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 18, 8, 1, 1, 2, 3, 7, 12, 27, 27, 16, 1, 1, 2, 3, 7, 12, 30, 43, 54, 16, 1, 1, 2, 3, 7, 12, 31, 55, 107, 81, 32, 1, 1, 2, 3, 7, 12, 31, 58, 141, 171, 162, 32, 1, 1, 2, 3, 7, 12, 31, 59, 159, 266, 427, 243, 64, 1, 1, 2, 3, 7, 12, 31, 59, 163, 312, 688, 683, 486, 64, 1

Offset: 1

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABC are equivalent, as are AAABB and BBBAA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a loop are equivalent, so for loops AAABCB = BAAABC = CBAAAB.

			The array begins at T(1,1):
1  1   1    1    1     1     1     1     1     1     1     1     1 ...
1  2   2    2    2     2     2     2     2     2     2     2     2 ...
1  2   3    3    3     3     3     3     3     3     3     3     3 ...
1  4   6    7    7     7     7     7     7     7     7     7     7 ...
1  4   9   11   12    12    12    12    12    12    12    12    12 ...
1  8  18   27   30    31    31    31    31    31    31    31    31 ...
1  8  27   43   55    58    59    59    59    59    59    59    59 ...
1 16  54  107  141   159   163   164   164   164   164   164   164 ...
1 16  81  171  266   312   334   338   339   339   339   339   339 ...
1 32 162  427  688   883   963   993   998   999   999   999   999 ...
1 32 243  683 1313  1774  2069  2169  2204  2209  2210  2210  2210 ...
1 64 486 1707 3407  5103  6119  6634  6789  6834  6840  6841  6841 ...
1 64 729 2731 6532 10368 13524 15080 15790 15975 16026 16032 16033 ...
a(n) are the terms of this array read by antidiagonals.
For T(4,3)=6, the achiral pattern rows are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA. The achiral pattern loops are AAAA, AAAB, AABB, ABAB, AABC, and ABAC.

Columns 1-6 are A057427, A016116, A182522, A305750, A305751, and A305752.

Columns converge to the right to A080107.

Ach[n_, k_] := Ach[n,k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] +
  Ach[n-2,k-1] + Ach[n-2,k-2]]; (* A304972 *)
Table[Sum[Ach[n, j], {j, 1, k - n + 1}], {k, 1, 15}, {n, 1, k}] // Flatten

T(n,k) = Sum_{j=0..k} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0 <= n <= 1 & n==k].

T(n,k) = Sum_{j=1..k} A304972(n,j).

A320936 Number of chiral pairs of color patterns (set partitions) for a row of length n using 6 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859, 1699012, 9808848, 57335124, 338073107, 2004955824, 11936998016, 71253827696, 426061036747, 2550545918300, 15280090686256, 91588065861292, 549159350303235, 3293482358956552, 19755007003402944

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056273 and A305752, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (16,-84,84,685,-2140,180,7200,-8244, -4176,11664,-5184).

Column 6 of A320751.

Cf. A056273 (oriented), A056325 (unoriented), A305752 (achiral).

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=6; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]
LinearRecurrence[{16, -84, 84, 685, -2140, 180, 7200, -8244, -4176, 11664, -5184}, {0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859}, 40]

concat([0,0], Vec(x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)) + O(x^40))) \\ Colin Barker, Nov 22 2018

a(n) = (A056273(n) - A305752(n))/2.
a(n) = A056273(n) - A056325(n).
a(n) = A056325(n) - A305752(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n) + A320529(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=6 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
From Colin Barker, Nov 22 2018: (Start)
G.f.: x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)).
a(n) = 16*a(n-1) - 84*a(n-2) + 84*a(n-3) + 685*a(n-4) - 2140*a(n-5) + 180*a(n-6) + 7200*a(n-7) - 8244*a(n-8) - 4176*a(n-9) + 11664*a(n-10) - 5184*a(n-11) for n>11.
(End)

A320746 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 34, 190, 996, 5070, 26454, 139484, 749742, 4082481, 22509626, 125231540, 702004040, 3958071545, 22423227634, 127524417922, 727617119592, 4163076477731, 23876455868772, 137228326265794, 790200053665362, 4557942281943078, 26331297198477874, 152331940294133402, 882422871962784662, 5117852332008063806, 29715786649820358328, 172720006045619486686, 1004904748993330281274, 5852047136464153694312

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056294 and A305752, which can be used in conjunction with the first formula.

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 6 of A320742.

Cf. A056294 (oriented), A056356 (unoriented), A305752 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=6; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

a(n) = (A056294(n) - A305752(n)) / 2 = A056294(n) - A056356(n) = A056356(n) - A305752(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=6 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n) + A320646(n).

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A056273 Word structures of length n using a 6-ary alphabet.

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A056325 Number of reversible string structures with n beads using a maximum of six different colors.

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A305749 T(n,k) is the number of achiral color patterns (set partitions) in a row or loop of length n with k or fewer colors (sets).

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A320936 Number of chiral pairs of color patterns (set partitions) for a row of length n using 6 or fewer colors (subsets).

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A320746 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets).

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