A176042 For odd n, this is the number of partitions of n*(n-1)/2 with all part sizes between 3 and n, inclusive. For even n, this is the number of partitions of n*(n-2)/2 with all part sizes between 3 and n, inclusive.

1, 1, 2, 5, 20, 42, 238, 511, 3311, 7423, 52273, 119739, 894950, 2087761, 16317275, 38616848, 312598141, 748492526, 6233339701, 15070028915, 128475055100, 313137867019, 2722580871465, 6681890398543, 59076953846060, 145856049509351, 1308316471338448

Offset

3,3

Comments

Number of different decompositions of complete graphs on n vertices into cycles (for odd n) or into cycles and a 1-factor (for n even) according to a conjecture of Brian Alspach's. This is verified for n <= 14.

Links

Alois P. Heinz, Table of n, a(n) for n = 3..500

B. Alspach, Research problems, Problem 3, Discrete Math 36 (1981), page 333.

Example

For n=3 we have 1+1+1 = 1+2 = 3 = n*(n-1)/2 of which only the final partition is counted.
For n=4 we have 1+1+1+1 = 1+1+2 = 2+2 = 1+3 = 4 = n*(n-2)/2 of which only the final partition is counted.
For n=5 we have n*(n-1)/2 = 10 and only 3+3+4 = 5+5 are counted.

Maple

a := proc (n) local i; option remember; if k::odd then coeff(series(1/(product(1-x^i, i = 3 .. n)), x, (1/2)*n*(n-1)+1), x, (1/2)*n*(n-1)) elif n::even then coeff(series(1/(product(1-x^i, i = 3 .. n)), x, (1/2)*n*(n-2)+1), x, (1/2)*n*(n-2)) end if end proc # Christopher Maitland, Jun 17 2010
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<3, 0,
      b(n, i-1)+b(n-i, min(n-i, i))))
    end:
a:= n-> `if`(n::odd, b(n*(n-1)/2, n), b(n*(n-2)/2, n)):
seq(a(n), n=3..35);  # Alois P. Heinz, Feb 21 2019

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 3, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := If[OddQ[n], b[n(n-1)/2, n], b[n(n-2)/2, n]];
a /@ Range[3, 35] (* Jean-François Alcover, Nov 21 2020, after Alois P. Heinz *)

Crossrefs

Sequence in context: A244087 A361134 A056726 * A185593 A136899 A136882

Adjacent sequences: A176039 A176040 A176041 * A176043 A176044 A176045

Keyword

nonn

Author

Christopher Maitland (c3053540(AT)uon.edu.au), Apr 07 2010

Extensions

Definition edited by Franklin T. Adams-Watters, Nov 17 2011

Status

approved