A094533 Number of one-element transitions among partitions of the integer n for labeled parts.

0, 0, 2, 8, 22, 48, 98, 178, 316, 524, 856, 1334, 2066, 3084, 4578, 6626, 9530, 13434, 18854, 26022, 35764, 48520, 65526, 87550, 116536, 153674, 201906, 263258, 342006, 441366, 567754, 726032, 925588, 1174010, 1484664, 1869072, 2346586, 2934044

Offset: 0

Author

Thomas Wieder, Jun 05 2004

Keywords

• nonn

Examples

			In the labeled case we have 22 one-element transitions among all partitions of n=4:
[1,1,1,1] -> [1,1,2] arises 6 times (the first 1 added to the second 1 gives 2, the first 1 added to the third 1 gives 2, the first 1 added to the fourth 1 gives 2, the second 1 added to the third 1 gives 2, the second 1 added to the fourth 1 gives 2, the third 1 added to the fourth 1 gives 2),
[1,1,2] -> [2,2] arises 1 times,
[1,1,2] -> [1,3] arises 2 times,
[2,2] -> [1,3] arises 1 times,
[1,3] -> [4] arises 1 time,
which gives 11 upwards transitions and 22 transitions in total if we include downwards transitions.
n=4: partition number p=1 is [1,1,1,1],
parts d(1,1)=1, d(2,1)=1 contribute 1,
parts d(1,1)=1, d(3,1)=1 contribute 1,
etc...
parts d(3,1)=1, d(4,1)=1 contribute 1,
(in total 6 contributions by [1,1,1,1]);
partition number p=2 is [1,1,2],
parts d(1,2)=1, d(2,2)=1 contribute 1,
parts d(1,2)=1, d(3,2)=2 contribute 1,
parts d(2,2)=1, d(3,2)=2 contribute 1;
partition number p=3 is [2,2],
parts d(1,3)=2, d(2,3)=2 contribute 1;
partition number p=4 is [1,3],
parts d(1,4)=1, d(2,4)=3 contribute 1;
partition number p=5 is [4],
part d(1,5)=4 contributes 0;
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
• Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol

Crossrefs

Cf. A093695.

Programs

Maple

main := proc(n::integer) local a,ndxp,ListOfPartitions,APartition,PartOfAPartition; with(combinat): ListOfPartitions:=partition(n); a:=0; for ndxp from 1 to nops(ListOfPartitions) do APartition := ListOfPartitions[ndxp]; a := a + nops(APartition)^2 - nops(APartition); end do; print("n, a(n):",n,a); end proc;
# second Maple program:
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, [1]] elif i<1 then [0, [0]]
    else f:= b(n, i-1); g:= `if`(i>n, [0, [0]], b(n-i, i));
         [f[1]+g[1], zip((x, y)-> x+y, f[2], [0, g[2][]], 0)]
      fi
    end:
a:= n-> (l-> add(l[t+1]*t*(t-1), t=1..nops(l)-1))(b(n$2)[2]):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 05 2012

Mathematica

a[n_] := Block[{p = IntegerPartitions[n], l = PartitionsP[n]}, Sum[ Length[p[[k]]]^2 - Length[p[[k]]], {k, l}]]; Table[ a[n], {n, 0, 37}] (* Robert G. Wilson v, Jul 13 2004, updated by Jean-François Alcover, Jan 29 2014 *)
Simplify@Table[SeriesCoefficient[(Log[1 - x]^2 - Log[1 - x] Log[x] + QPolyGamma[1, x] (2 Log[1 - x] - Log[x] + QPolyGamma[1, x]) + QPolyGamma[1, 1, x])/(QPochhammer[x] Log[x]^2), {x, 0, n}], {n, 0, 40}] (* Vladimir Reshetnikov, Nov 21 2016 *)
Simplify@Table[SeriesCoefficient[2 q^2/QPochhammer[q + a, q], {a, 0, 2}, {q, 0, n}], {n, 0, 40}] (* Vladimir Reshetnikov, Nov 22 2016 *)

Formula

a(n) = Sum_p=1^P(n) Sum_i=1^D(p) Sum_j=i^D(p) 1 [subject to: d(i, p) <= d(j, p) ]; P(n) = number of partitions of n, D(p) = number of parts in partition p, d(i, p) and d(j, p) = parts number i and j in partition p of integer n.

a(n) = Sum_i=1^P(n) p(i, n)^2 - p(i, n), where P(n) is the number of integer partitions of n and p(i, n) is the number of parts of the i-th partition of n.

G.f.: (log(1-x)^2 - log(1-x)*log(x) + psi_x(1)*(2*log(1-x) - log(x) + psi_x(1)) + psi^1_x(1))/((x; x)inf * log(x)^2), where psi_q(z) is the q-digamma function, psi^1_q(z) is the q-trigamma function, and (a; q)_inf is the q-Pochhammer symbol (the Euler function). To get this g.f., take the derivative (d/da)^2 (x^2/(a; x)_inf) and let a = x. - _Vladimir Reshetnikov, Nov 21 2016

Extensions

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