A058884 - OEIS

%I #40 May 19 2023 18:49:43

%S -1,0,0,1,2,5,8,15,23,37,55,83,118,171,238,332,453,618,827,1107,1460,

%T 1922,2504,3253,4188,5380,6860,8722,11024,13895,17421,21787,27122,

%U 33677,41653,51390,63179,77496,94755,115600,140632,170725,206717,249804,301151,362367,435077,521439,623674,744695

%N Partial sums of the partition function (A000041), with the last term subtracted. Also the sum of the row of the character table for S_n corresponding to the partition n-1,1 for n>1. Also the sum over all partitions lambda of n of one less than the number of 1's in lambda.

%C For n>=1 number of up-steps in all partitions of n (represented as weakly increasing lists), see example. - _Joerg Arndt_, Sep 03 2014

%H Andrew Howroyd, <a href="/A058884/b058884.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Archibald/arch3.html">Inversions and Parity in Compositions of Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.

%H Anders Claesson, Atli Fannar Franklín, and Einar Steingrímsson, <a href="https://arxiv.org/abs/2305.09457">Permutations with few inversions</a>, arXiv:2305.09457 [math.CO], 2023.

%H S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, <a href="https://www.researchgate.net/publication/228671252_Inversions_in_compositions_of_integers">Inversions in Compositions of Integers</a>, Quaestiones Mathematicae 34 (2011), 187-202.

%F From _Andrew Howroyd_, Apr 21 2023: (Start)

%F a(n) = A000070(n-1) - A000041(n) for n > 0.

%F G.f.: (2*x - 1)*P(x)/(1 - x) where P(x) is the g.f. of A000041. (End)

%e a(6) = 8 because the 11 partitions of 6

%e 01:  [ 1 1 1 1 1 1 ]

%e 02:  [ 1 1 1 1 2 ]

%e 03:  [ 1 1 1 3 ]

%e 04:  [ 1 1 2 2 ]

%e 05:  [ 1 1 4 ]

%e 06:  [ 1 2 3 ]

%e 07:  [ 1 5 ]

%e 08:  [ 2 2 2 ]

%e 09:  [ 2 4 ]

%e 10:  [ 3 3 ]

%e 11:  [ 6 ]

%e contain 0+1+1+1+1+2+1+0+1+0+0 = 8 up-steps. - _Joerg Arndt_, Sep 03 2014

%p a:= proc(n) uses combinat; add(numbpart(k), k=0..n-1)-numbpart(n) end:

%p seq(a(n), n=0..49);

%t p[n_] := IntegerPartitions[n]; l[n_] := Length[p[n]]; Table[Count[Flatten[p[n]], 1] - l[n], {n, 0, 30}] (* _Clark Kimberling_, Mar 08 2012 *)

%o (PARI) a(n) = {sum(k=0, n-1, numbpart(k)) - numbpart(n)} \\ _Andrew Howroyd_, Apr 21 2023

%o (PARI) Vec((2*x - 1)/(1 - x)/eta(x + O(x^51))) \\ _Andrew Howroyd_, Apr 21 2023

%Y Cf. A000041, A000070.

%Y Cf. A218074 (up-steps in partitions into distinct parts).

%K sign,easy

%O 0,5

%A _Edward Early_, Jan 08 2001

%E More terms from _James A. Sellers_, Sep 28 2001