A079816 Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=5, I={1}.

1, 1, 1, 2, 4, 7, 12, 20, 34, 59, 102, 175, 300, 515, 885, 1521, 2613, 4488, 7709, 13243, 22750, 39081, 67134, 115324, 198107, 340315, 584604, 1004250, 1725130, 2963480, 5090756, 8745055, 15022519, 25806135, 44330556, 76152366, 130816831

Offset: 0

Author

Vladimir Baltic, Feb 19 2003

Keywords

• nonn

Comments

Number of compositions (ordered partitions) of n into elements of the set {1,3,4,5,6}.

a(n+1) is the number of multus bitstrings of length n with no runs of 6 ones. - Steven Finch, Mar 25 2020

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135
• Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,1,1,1).

Crossrefs

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

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x-x^3-x^4-x^5-x^6) )); // G. C. Greubel, Dec 12 2023

Mathematica

LinearRecurrence[{1,0,1,1,1,1}, {1,1,1,2,4,7}, 51] (* G. C. Greubel, Dec 12 2023 *)

SageMath

def A079816_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x-x^3-x^4-x^5-x^6) ).list()
A079816_list(50) # G. C. Greubel, Dec 12 2023

Formula

Recurrence: a(n) = a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6).

G.f.: 1/(1-x-x^3-x^4-x^5-x^6).