A323936 Isomers of deoxyalditols.

1, 1, 3, 6, 15, 32, 74, 160, 356, 768, 1672, 3584, 7696, 16384, 34848, 73728, 155712, 327680, 688256, 1441792, 3014912, 6291456, 13107712, 27262976, 56624128, 117440512, 243271680, 503316480, 1040191488, 2147483648, 4429193216, 9126805504

Offset

1,3

Comments

From Travis Scott, Feb 15 2023: (Start)

Let b(i) give the i-th bit of a string indexed {0...j-1}, let s > 1 divide j, define (s-)seam as a decimation b(i == r (mod j/s)), and let n count carbon atoms. Then the stable deoxy-n-itol isomers of Nemba and Fah enumerate as bit strings of length 2n and Hamming weight h = n+1, up to reversal, with b(0) = b(2n-1) = 1, and such that a 2-seam colored 00 destabilizes a string. [Ex. 111100 fails because its last bit is 0 and 101101 fails because it decimates 1 (mod 3) to 00.]

The seam 0 (mod n) may not be 01 and the seam n-1 (mod n) may not be 10; otherwise, 2-seams permute freely as compositions of n+1 into n parts using one kind of 2 (11) and two kinds of 1 (01 10). The number of stable strings follows by inex [exclude the two restrictions disjointly and include them jointly: A001787(n) - 2*A001787(n-1) + A001787(max(n-2, 0))]: combining terms gives A045623(n-1).

Now if n is even, h is odd and no palindromes may be colored, while if n is odd, palindromy induces a monochromatic seam (n-1)/2 (mod n) which may only be colored 11. Stable strings invariant to reversal are thus palindromes of length 2n == 2 (mod 4) with min(n, 2) impermutable bit pairs [their endpoints must also be colored 11; the pairs coincide when n=1] so that, in this case, the inex of curated wreath products exemplified at Table I of NF95 reduces to (A045623(n-1) + A011782((n-1)/2)(n mod 2))/2. (End)

Links

Table of n, a(n) for n=1..32.

K. Balasubramanian, Enumeration of stable stereo and position isomers of polysubstituted alcohols, Ann. N.Y. Acad. Sci. 319.1 (1979), pp. 33-36.

R. M. Nemba and M. Fah, On the application of sieve formula to the enumeration of stable stereo and position isomers of deoxyaldytols [sic], Tetrahedron 51.13 (1995), pp. 3831-3840. See Table II.

Formula

a(n) = 2^max(n-4, -2)(n+2) + 2^((n-5)/2)(n mod 2).

a(2^n - 2) = 2^(2^n + n-6).

Example

From Travis Scott, Feb 15 2023: (Start)
The a(5)=15 stable strings counting deoxypentitols, grouped by position-isomer equivalence [see B79, Fig. 4, cit. NF95 ante Table II]:
(A) 1000011111, 1000101111, 1001011101, 1001101101, 1010011011, 1010101011, 1100010111, 1100100111;
(B) 1001001111, 1011001011, 1100001111, 1101000111;
(C) 1010001111, 1011001101, 1110000111. (End)

Mathematica

Table[(n+2)2^Max[n-4, -2]+Mod[n, 2]2^((n-5)/2), {n, 32}]

Crossrefs

Cf. A001787, A045623, A011782.

Sequence in context: A316219 A034740 A367293 * A305839 A322110 A232973

Adjacent sequences: A323933 A323934 A323935 * A323937 A323938 A323939

Keyword

nonn

Author

N. J. A. Sloane, Feb 09 2019

Extensions

a(1)-a(2) prepended, more terms added by Travis Scott, Feb 15 2023

Status

approved