A079972 Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=4, I={1,2}.

1, 1, 1, 1, 2, 4, 6, 8, 11, 17, 27, 41, 60, 88, 132, 200, 301, 449, 669, 1001, 1502, 2252, 3370, 5040, 7543, 11297, 16919, 25329, 37912, 56752, 84968, 127216, 190457, 285121, 426841, 639025, 956698, 1432276, 2144238, 3210104, 4805827, 7194801

Offset: 0

Author

Vladimir Baltic, Feb 17 2003

Keywords

• nonn

Comments

Number of compositions (ordered partitions) of n into elements of the set {1,4,5}.

a(n+3) is the number of length-n binary words with no substring 1x1 of 1xy1 (that is, no 1's occur with distance two or three), see fxtbook link. - Joerg Arndt, May 27 2011

References

• D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..5708
• Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See p. 18.
• Said Amrouche and Hacène Belbachir, Unimodality and linear recurrences associated with rays in the Delannoy triangle, Turkish Journal of Mathematics (2019) Vol. 44, 118-130.
• Joerg Arndt, Matters Computational (The Fxtbook), section 14.10.3, p. 322
• Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,1).

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Programs

Mathematica

CoefficientList[Series[1/(1 - x - x^4 - x^5), {x, 0, 41}], x] (* Michael De Vlieger, Feb 03 2020 *)

Maxima

a(n):=sum(sum(binomial(k,j)*binomial(j,n-k-3*j),j,floor((n-k)/4),floor((n-k)/3)),k,0,n); /* Vladimir Kruchinin, May 26 2011 */

Formula

a(n) = a(n-1) + a(n-4) + a(n-5).

G.f.: 1/(1-x-x^4-x^5).

a(n) = Sum_{k=0..n} Sum_{j=floor((n-k)/4)..floor((n-k)/3)} binomial(k,j)*binomial(j,n-k-3*j). - Vladimir Kruchinin, May 26 2011