A056272 -id:A056272 - cp's OEIS frontend

A082951 Number of primitive (aperiodic) word structures of length n using an infinite alphabet.

1, 1, 1, 4, 13, 51, 197, 876, 4125, 21142, 115922, 678569, 4213381, 27644436, 190898444, 1382958489, 10480138007, 82864869803, 682076784814, 5832742205056, 51724158119384, 474869816155870, 4506715737768752, 44152005855084345, 445958869290587567

Offset: 0

Vadim Ponomarenko (vadim123(AT)gmail.com), May 26 2003

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

Row sums of triangle A137651. - Gary W. Adamson, Feb 01 2008

			There are A000110(3)=5 word structures of length 3: aaa, aab, aba, abb, abc. The first consists of 3 copies of a word of length 1; the other 4 are primitive. So a(3)=4.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000110, A056277, A056272, A056275, A056274, A056278.

Cf. A137651.

with(combinat,bell): with(numtheory): newb := proc(n) local s,i; s := 0; for i in divisors(n) do s := s+bell(i)*mobius(n/i): end do: end proc;
# second Maple program:
with(combinat): with(numtheory):
a:= proc(n) option remember;
      bell(n)-add(a(d), d=divisors(n) minus {n})
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 23 2015

a[n_] := DivisorSum[n, BellB[#] MoebiusMu[n/#]&]; a[0]=1; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 23 2017 *)

a(n) = sum mu(c)*A000110(d) over all cd=n; equivalently, A000110(n) = sum a(k), where the sum is over all k|n.

1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) is the g.f. of A000110. - Ilya Gutkovskiy, Mar 05 2018

More terms from Alois P. Heinz, Jan 23 2015

A320935 Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 400, 1852, 8868, 42892, 210346, 1038034, 5150110, 25623486, 127740880, 637539592, 3184224728, 15910524632, 79520923966, 397508610454, 1987255480650, 9935410066186, 49674450471460, 248364429410332, 1241798688445588, 6208922948527572, 31044403310614786

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 A056272 and A305751, 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.

Index entries for linear recurrences with constant coefficients, signature (11,-34,-16,247,-317,-200,610,-300).

Column 5 of A320751.

Cf. A056272 (oriented), A056324 (unoriented), A305751 (achiral).

LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {0, 0, 1, 4, 20, 86, 400, 1852}, 40] (* or *)
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=5; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]

a(n) = (A056272(n) - A305751(n))/2.
a(n) = A056272(n) - A056324(n).
a(n) = A056324(n) - A305751(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=5 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)).
G.f.: x^3*(1 - 7*x + 10*x^2 + 18*x^3 - 49*x^4 + 25*x^5)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 5*x)*(1 - 5*x^2)*(1 - 2*x^2)). - Bruno Berselli, Oct 31 2018

A056276 Number of primitive (aperiodic) word structures of length n using a 5-ary alphabet.

Original entry on oeis.org

1, 1, 4, 13, 51, 196, 854, 3830, 17997, 86419, 422004, 2079260, 10306751, 51263086, 255514299, 1275160060, 6368612301, 31821454413, 159042661904, 795019250650, 3974515029793, 19870830290476, 99348921288654, 496728909635860, 2483597478617750, 12417846151236799

Offset: 1

Marks R. Nester

nonn

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

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]

Cf. A008683, A054720.

a(n) = Sum_{d|n} mu(d)*A056272(n/d).

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A082951 Number of primitive (aperiodic) word structures of length n using an infinite alphabet.

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A320935 Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).

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A056276 Number of primitive (aperiodic) word structures of length n using a 5-ary alphabet.

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