A058884 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.
-1, 0, 0, 1, 2, 5, 8, 15, 23, 37, 55, 83, 118, 171, 238, 332, 453, 618, 827, 1107, 1460, 1922, 2504, 3253, 4188, 5380, 6860, 8722, 11024, 13895, 17421, 21787, 27122, 33677, 41653, 51390, 63179, 77496, 94755, 115600, 140632, 170725, 206717, 249804, 301151, 362367, 435077, 521439, 623674, 744695
Offset: 0
Examples
a(6) = 8 because the 11 partitions of 6 01: [ 1 1 1 1 1 1 ] 02: [ 1 1 1 1 2 ] 03: [ 1 1 1 3 ] 04: [ 1 1 2 2 ] 05: [ 1 1 4 ] 06: [ 1 2 3 ] 07: [ 1 5 ] 08: [ 2 2 2 ] 09: [ 2 4 ] 10: [ 3 3 ] 11: [ 6 ] contain 0+1+1+1+1+2+1+0+1+0+0 = 8 up-steps. - _Joerg Arndt_, Sep 03 2014
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1000
- M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
- Anders Claesson, Atli Fannar Franklín, and Einar Steingrímsson, Permutations with few inversions, arXiv:2305.09457 [math.CO], 2023.
- S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, Inversions in Compositions of Integers, Quaestiones Mathematicae 34 (2011), 187-202.
Programs
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Maple
a:= proc(n) uses combinat; add(numbpart(k), k=0..n-1)-numbpart(n) end: seq(a(n), n=0..49);
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Mathematica
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 *) -
PARI
a(n) = {sum(k=0, n-1, numbpart(k)) - numbpart(n)} \\ Andrew Howroyd, Apr 21 2023 -
PARI
Vec((2*x - 1)/(1 - x)/eta(x + O(x^51))) \\ Andrew Howroyd, Apr 21 2023
Formula
From Andrew Howroyd, Apr 21 2023: (Start)
G.f.: (2*x - 1)*P(x)/(1 - x) where P(x) is the g.f. of A000041. (End)
Extensions
More terms from James Sellers, Sep 28 2001
Comments