A007075 Number of irreducible positions of size n in Montreal solitaire. (Formerly M1441)

1, 2, 5, 13, 35, 95, 260, 714, 1965, 5415, 14934, 41206, 113730, 313958, 866801, 2393315, 6608473, 18248017, 50389350, 139144906, 384237186, 1061044865, 2930013158, 8091077148, 22343115337, 61699480866, 170380367189, 470497972866

Offset

1,2

Comments

a(n) is also the number of indecomposable permutations with exactly n inversions; there is one indecomposable permutation with no inversions. - David Bevan, Dec 19 2017

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

David Bevan, Table of n, a(n) for n = 1..1000

C. Cannings, J. Haigh, Montreal solitaire, J. Combin. Theory Ser. A 60 (1992), no. 1, 50-66.

Formula

a(n) = d(n, 1) where d(n, k) is defined in A007046. - Sean A. Irvine, Oct 06 2017

The ordinary generating function is f(1), where f(v) satisfies the functional equation f(v) = v*(1 + f(1 + x*v) - f(1)). The variable x marks inversions and v marks left-to-right minima. - David Bevan, Dec 19 2017

Example

a(3) = 5; five indecomposable permutations have three inversions: 321, 2341, 2413, 3142, 4123. - David Bevan, Dec 19 2017

Mathematica

r[1, 1]=1; r[_, 0]:=0; r[n_, k_]:=r[n, k]=Sum[r[n-k, j]Binomial[j+1, k], {j, k-1, (Sqrt[8(n-k)+1]-1)/2}]; a[n_]:=Sum[r[n, k], {k, (Sqrt[8n+1]-1)/2}]; Array[a, 20] (* David Bevan, Dec 19 2017 *)

Crossrefs

Cf. A007046, A007048, A007049, A007050, A007076.

Sequence in context: A057960 A339479 A227045 * A000107 A366088 A370841

Adjacent sequences: A007072 A007073 A007074 * A007076 A007077 A007078

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein

Extensions

More terms from Sean A. Irvine, Oct 06 2017

Status

approved