0,2

These objects are quite similar to "meanders". They were introduced by Philippe di Francesco, with the hope that people in combinatorics would be able to prove some facts on this model, which is a kind of "meander model" with local constrains (and not globally, unlike meanders). Any asymptotic equivalent, or any closed-form formula would be an interesting result. Only trivial upper bounds like 2/Pi 4^n/n^3 are known. KPZ formula from CFT gives an hint on what could be the critical exponent of this sequence. A good computational scheme for getting for terms is also a challenge.

C. Banderier, Parity-constrained-shuffle of Dyck Words.

a(4)=228. Indeed, here are the 228 words of length 8:

[cdccddcd, cccdcddd, cdcdcdcd, cdcdccdd, cccdddcd, cccddcdd, cdccdcdd, ccdcddcd, ccdcdcdd, ccdccddd, ccddcdcd, cdcccddd, ccddccdd, ccccdddd, ccddabcd, ccddacdb, ccddcdab, ccddcabd, ccdabdcd, ccabddcd, cabcddcd, cacdbdcd, abcccddd, acccdddb, abccdcdd, accdcddb, cdcdabcd, cdcdacdb, cdcdcdab, cdcdcabd, cdcabdcd, cdacdcdb, cdacdbcd, cdabcdcd, cabdcdcd, cabdccdd, cdabccdd, cdaccddb, ccabdcdd, ccdabcdd, ccdacdbd, ccdcabdd, ccdcddab, ccdcdabd, cabcdcdd, cacdbcdd, cacdcdbd, cdccabdd, cdccddab, cdccdabd, caccddbd, cabccddd, ccacdbdd, ccabcddd, cccabddd, cccdabdd,

cccdddab, cccddabd, abccddcd, accddbcd, accddcdb, abcdccdd, acdbccdd, acdccddb, abcdcdcd, acdbcdcd, acdcdcdb, acdcdbcd, cdcabcdd, cdcacdbd, cdcdaabb, cdaacdbb, caacdbbd, caabbcdd, caabcdbd, cacdabbd, ccddaabb, ccaabbdd, ccaaddbb, abcdcdab, acdbabcd, acdbacdb, acdbcdab, acdcdbab, acdbcabd, acdcdabb, acdabbcd, acdabcdb, acdacdbb, acdcabdb, acabdbcd, abcabdcd, abcdcabd, cdacdbab, cdabcabd, cdababcd, cdabacdb, cdabcdab, cababdcd, ababcdcd, abacdbcd, abacdcdb, abcdabcd, abcdacdb, accbaddb, accddbab, cabdcdab, cabdcabd, acabdcdb, aacdbcdb, aacdcdbb, aabbcdcd,

aabcdbcd, aabcdcdb, cababcdd, cabacdbd, cabcabdd, aacdbbcd, caabbdcd, cabcddab, cabcdabd, ccababdd, ccabddab, ccabdabd, cacdbdab, cacdbabd, accabddb, aaccddbb, aabbccdd, aabccddb, accdabdb, accddabb, acacdbdb, acabcddb, cdcabdab, cdcababd, cdcdabab, cabdabcd, cabdacdb, ccdaabbd, cacabdbd, aaccbbdd, abaccddb, ccdabdab, ccdababd, ccddabab, abcabcdd, abcacdbd, abccabdd, abccddab, abccdabd, ababccdd, caadcbbd, cdacdabb, cdaabbcd, cdaabcdb, cdacabdb, cdcaabbd, acdbabab, caababbd, cabdaabb, abcabdab, abcababd, aabcabdb, aabacdbb, cdaaabbb, caabbdab, caabbabd, cdaabbab, acababdb, acabdabb, acdababb, aabcdabb, aababbcd, aacdbabb, aababcdb, abcdabab, abacdbab, ababcabd, abababcd, cabaabbd, caaabbbd, aacdabbb, aaacdbbb, aaabbbcd, aacabdbb, aaabbcdb, aaabcdbb, abacdabb, abaabbcd, abaabcdb, abaacdbb, abacabdb, abcaabbd, abcdaabb, acdbaabb, cdaababb, cababdab, cabababd, cabdabab, cdababab, cdabaabb, acdaabbb, acaabbdb, acdabbab, aacdbbab, acabdbab, aabcdbab, aabbcabd, aabbabcd, aabbacdb,

aabbcdab, ababacdb, ababcdab, abaabbab, aaababbb, abababab, ababaabb, aaabbbab, aaabbabb, abaababb, aababbab, aabababb, aabaabbb, aabbabab, abaaabbb, aabbaabb, aaaabbbb]

Sequence in context: A052701 A085485 A089603 this_sequence A055882 A002301 A000828

Adjacent sequences: A116453 A116454 A116455 this_sequence A116457 A116458 A116459

nonn

Cyril Banderier (cyril.banderier(AT)lipn.univ-paris13.fr), Mar 16 2006