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A210109 Number of 3-divided binary sequences (or words) of length n. 13
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 (list; graph; refs; listen; history; text; internal format)



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.


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. Sch├╝tzenberger, 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.


Table of n, a(n) for n=1..36.

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).


Is there a formula or recurrence?


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.











(Python) # 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


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.

Sequence in context: A011757 A121963 A041159 * A034546 A281584 A230315

Adjacent sequences:  A210106 A210107 A210108 * A210110 A210111 A210112




N. J. A. Sloane, Mar 17 2012


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

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



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Last modified September 26 06:50 EDT 2022. Contains 356987 sequences. (Running on oeis4.)