A209919 -id:A209919 - cp's OEIS frontend

A210109 Number of 3-divided binary sequences (or words) of length n.

0, 0, 0, 2, 7, 23, 54, 132, 290, 634, 1342, 2834, 5868, 12140, 24899, 50929, 103735, 210901, 427623, 865910, 1750505, 3535098, 7131321, 14374647, 28952661, 58280123, 117248217, 235770302, 473897980, 952183214, 1912535827, 3840345963, 7709282937, 15472242645, 31045402788, 62280978042

Offset: 1

N. J. A. Sloane, Mar 17 2012

nonn

A binary sequence (or word) W of length n is 3-divided if it can be written as a concatenation W = XYZ such that XYZ is strictly earlier in lexicographic order than any of the five permutations  XZY, ZYX, YXZ, YZX, ZXY.
More generally, fix an alphabet A = {0,1,2,...,a-1}.
Define lexicographic order on words over A in the obvious way: for single letters, i < j is the natural order; for words U, V, we set U < V iff u_i < v_i at the first place where they differ; also U < UV if V is nonempty, etc.
Then a word U over A is "k-divided over A" if it can be written as U = X_1 X_2 ... X_k in such a way that X is strictly less in lexicographic order than X_pi_1 X_pi_2 ... X_pi_k for all nontrivial permutations pi of [1..k].
All 2^n binary words are 1-divided. For 2-divided words see A209970.
"k-divisible" would sound better to me than "k-divided", but I have followed Lothaire and Pirillo-Varricchio in using the latter term. Neither reference gives this sequence.

			The two 3-divisible binary words of length 4 and the seven of length 5 are as follows. The periods indicate the division w = x.y.z. For example, 0.01.1 is 3-divided since 0011 < all of 0101, 1010, 0101, 1001, 0110.
0.01.1
0.10.1
0.001.1
0.010.1
0.01.10
0.01.11
0.100.1
0.10.11
0.110.1

M. Lothaire, Combinatorics on words. A collective work by Dominique Perrin, Jean Berstel, Christian Choffrut, Robert Cori, Dominique Foata, Jean Eric Pin, Guiseppe Pirillo, Christophe Reutenauer, Marcel-P. Schützenberger, Jacques Sakarovitch and Imre Simon. With a foreword by Roger Lyndon. Edited and with a preface by Perrin. Encyclopedia of Mathematics and its Applications, 17. Addison-Wesley Publishing Co., Reading, Mass., 1983. xix+238 pp. ISBN: 0-201-13516-7, MR0675953 (84g:05002). See p. 144.

Michael S. Branicky, Python program using bitwise operations
Giuseppe Pirillo and Stefano Varricchio, Some combinatorial properties of infinite words and applications to semigroup theory. Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993). Discrete Math. 153 (1996), no. 1-3, pages 239-251, MR1394958 (98f:05018).

Number of k-divided words of length n over alphabet of size A:
A=2, k=2,3,4,5: A209970 (and A209919, A000031, A001037), A210109 (and A210145), A210321, A210322;
A=3, k=2,3,4,5: A210323 (and A001867, A027376), A210324, A210325, A210326;
A=4, k=2,3,4: A210424 (and A001868, A027377), A210425, A210426.

# see link for faster, bit-based version
from itertools import product
def is3div(b):
    for i in range(1, len(b)-1):
        for j in range(i+1, len(b)):
            X, Y, Z = b[:i], b[i:j], b[j:]
            if all(b < bp for bp in [Z+Y+X, Z+X+Y, Y+Z+X, Y+X+Z, X+Z+Y]):
                return True
    return False
def a(n): return sum(is3div("".join(b)) for b in product("01", repeat=n))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Aug 27 2021

Is there a formula or recurrence?

Values confirmed and a(30)-a(31) by David Applegate, Mar 19 2012

a(32)-a(36) from Michael S. Branicky, Aug 27 2021

A209970 a(n) = 2^n - A000031(n).

Original entry on oeis.org

0, 0, 1, 4, 10, 24, 50, 108, 220, 452, 916, 1860, 3744, 7560, 15202, 30576, 61420, 123360, 247542, 496692, 996088, 1997272, 4003558, 8023884, 16077964, 32212248, 64527436, 129246660, 258847876, 518358120, 1037949256, 2078209980, 4160747500, 8329633416, 16674575056, 33378031536

Offset: 0

David Applegate and N. J. A. Sloane, Mar 17 2012

nonn

a(n) is also the number of 2-divided binary words of length n (see A210109 for definition, see A209919 for further details).
This is a special case of a more general result: Let A={0,1,...,s-1} be an alphabet of size s. Let A* = set of words over A. Let < denote lexicographic order on A*. Let f be the morphism on A* defined by i -> s-i for i in A.
Theorem: Let d(n) be the number of 2-divided words in A* of length n, and let b(n) be the number of rotationally inequivalent necklaces with n beads each in A. Then d(n)+b(n)=s^n.
Proof: Let w in A* have length n. If w is <= all of its cyclic shifts then w contributes to the b(n) count. Otherwise w = uv with vu < uv. But then f(w)=f(u)f(v) with f(u)f(v) < f(v)f(u) is 2-divided, and w contributes to the count in d(n). QED
Cor.: A000031(n) + A209970(n) = 2^n, A001867(n) + A210323(n) = 3^n, A001868(n) + A210424(n) = 4^n.

Cf. A000031, A210109, A209919.

A210323 Number of 2-divided words of length n over a 3-letter alphabet.

Original entry on oeis.org

0, 3, 16, 57, 192, 599, 1872, 5727, 17488, 53115, 161040, 487073, 1471680, 4441167, 13392272, 40355877, 121543680, 365895947, 1101089808, 3312442185, 9962240928, 29954639751, 90049997136, 270661616363, 813397065024, 2444101696683, 7343167947040, 22059763982001, 66263812628160

Offset: 1

N. J. A. Sloane, Mar 20 2012

nonn

See A210109 for further information.
It appears that A027376 gives the number of 2-divided words that have a unique division into two parts. - David Scambler, Mar 21 2012
Row sums of the following irregular triangle W(n,k) which shows how many words of length n over a 3-letter alphabet are 2-divided in k>=1 different ways (3-letter analog of A209919):
3;
8, 8;
18, 21, 18;
48, 48, 48, 48;
116, 124, 119, 124, 116;
312, 312, 312, 312, 312, 312;
810, 828, 810, 831, 810, 828, 810;
2184, 2184, 2192, 2184, 2184, 2192, 2184, 2184;
5880, 5928, 5880, 5928, 5883, 5928, 5880, 5928, 5880;
First column of the following triangle D(n,k) which shows how many words of length n over a 3-letter alphabet are k-divided:
3;
16, 1;
57, 16, 0;
192, 78, 6, 0;
599, 324, 56, 0, 0;
1872, 1141, 343, 15, 0, 0;
5727, 3885, 1534, 166, 0, 0, 0;
17488, 12630, 6067, 1135, 20, 0, 0, 0;
53115, 40315, 22162, 5865, 351, 0, 0, 0, 0;
161040, 126604, ...
- R. J. Mathar, Mar 25 2012
Speculation: W(2n+1,2)=W(2n+1,1) and W(2n,2) = W(2n,1)+W(n,1). W(3n+1,3)=W(3n+1,1); W(3n+2,3)=W(3n+1,1); W(3n,3) = W(3n,1)+W(n,1). - R. J. Mathar, Mar 27 2012

Cf. A210109, A209970, A001867.

a(n) = 3^n - A001867(n) (see A209970 for proof).

a(1)-a(12) computed by David Scambler, Mar 19 2012; a(13) onwards from N. J. A. Sloane, Mar 20 2012

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A210109 Number of 3-divided binary sequences (or words) of length n.

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A209970 a(n) = 2^n - A000031(n).

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A210323 Number of 2-divided words of length n over a 3-letter alphabet.

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