A319420 Irregular triangle read by rows: row n lists the cuts-resistances of the 2^n binary vectors of length n.

0, 1, 1, 2, 1, 1, 2, 3, 2, 1, 2, 2, 1, 2, 3, 4, 3, 2, 2, 2, 1, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 5, 4, 3, 3, 3, 2, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 3, 3, 4, 5

Offset: 0

Author

N. J. A. Sloane, Sep 22 2018

Keywords

• nonn
• tabf
• more

Comments

The cuts-resistance of a vector is defined in A319416. The 2^n vectors of length n are taken in lexicographic order.

Note that here the vectors can begin with either 0 or 1, whereas in A319416 only vectors beginning with 1 are considered (since there we are considering binary representations of numbers).

Conjecture: The row sums, halved, appear to match A189391.

Examples

			Triangle begins:
0,
1,1,
2,1,1,2,
3,2,1,2,2,1,2,3,
4,3,2,2,2,1,2,3,3,2,1,2,2,2,3,4,
5,4,3,3,3,2,2,3,3,2,1,2,2,2,3,4,4,3,2,2,2,1,2,3,3,2,2,2,3,3,3,4,5,
...
		

Links

• Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. See table on page 4.

Crossrefs

Keeping the first digit gives A319416.

Positions of 1's are the terms > 1 of A061547 and A086893, all minus 1.

The version for runs-resistance is A329870.

Compositions counted by cuts-resistance are A329861.

Binary words counted by cuts-resistance are A319421 or A329860.

Cf. A000975, A027383, A189391, A318921, A318928, A319411, A329767, A329862, A329865.

Programs

Mathematica

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[degdep[Rest[IntegerDigits[n,2]]],{n,0,50}] (* Gus Wiseman, Nov 25 2019 *)