A096586 Number of one-element transitions among all integer partitions of the integers from m=0 to m=n in the unlabeled case.

0, 2, 8, 20, 44, 86, 158, 274, 458, 738, 1160, 1778, 2674, 3948, 5744, 8236, 11670, 16344, 22664, 31126, 42390, 57260, 76790, 102260, 135320, 177976, 232778, 302814, 391972, 504948, 647592, 826956, 1051750, 1332438, 1681856, 2115376, 2651726

Offset: 0

Author

Thomas Wieder, Jul 02 2004

Keywords

• nonn

Comments

We set A096586(0) = 0.

Examples

			a(5) = 2*43 = 86 because:
11 -> 2, 111 -> 12, 12 -> 3, 1111 -> 112, 112 -> 13, 112 -> 22,
13 -> 22, 13 -> 4, 11111 -> 1112, 1112 -> 122, 1112 -> 113, 122 -> 23,
122 -> 113, 113 -> 23, 113 -> 14, 23 -> 14, 14 -> 5,
0 -> 1,
1 -> 11, 1 -> 2, 11 -> 111, 11 -> 12, 2 -> 12, 2 -> 3, 111 -> 1111,
111 -> 112, 12 -> 112, 12 -> 13, 12 -> 22, 3 -> 13, 3 -> 4,
1111 -> 11111, 1111 -> 1112, 112 -> 1112, 112 -> 113, 112 -> 122,
13 -> 113, 13 -> 14, 13 -> 23, 22 -> 23, 22 -> 122, 4 -> 14, 4 -> 5,
which gives 43 transitions and (counting upwards and downwards transitions) we have 2*43 = 86 = A096586(5).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A093695, A000070.

Programs

Mathematica

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) a[0] = 0; a[n_] := Block[{p = Partitions[n + 1], l = PartitionsP[n + 1]}, Sum[ Length[ Union[ p[[k]] ]]^2 - Length[ Union[ p[[k]] ]], {k, l}]]; b = CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x]; f[n_] := Sum[a[k] + 2b[[k]], {k, n}] - 1; Table[ f[n], {n, 36}] (* Robert G. Wilson v, Jul 13 2004 *)

Formula

A096586(n) = Sum_k=0^n A093695(k) + 2 * Sum_l=0^(n-1) A000070(l).

Extensions

More terms from Robert G. Wilson v, Jul 13 2004