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%I #139 May 20 2024 11:24:48
%S 0,1,1,3,4,8,11,19,26,41,56,83,112,160,213,295,389,526,686,911,1176,
%T 1538,1968,2540,3223,4115,5181,6551,8191,10269,12756,15873,19598,
%U 24222,29741,36532,44624,54509,66261,80524,97446,117862,142029,171036,205290,246211
%N Number of 2's in all partitions of n.
%C Also number of partitions of n-1 with a distinguished part different from all the others. [Comment corrected by _Emeric Deutsch_, Aug 13 2008]
%C In general the number of times that j appears in the partitions of n equals Sum_{k<n, k = n (mod j)} P(k). In particular this gives a formula for a(n), A024787, ..., A024794, for j = 2,...,10; it generalizes the formula given for A000070 for j=1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
%C Equals row sums of triangle A173238. - _Gary W. Adamson_, Feb 13 2010
%C The sums of two successive terms give A000070. - _Omar E. Pol_, Jul 12 2012
%C a(n) is also the difference between the sum of second largest and the sum of third largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - _Omar E. Pol_, Oct 25 2012
%C Number of singletons in all partitions of n-1. A singleton in a partition is a part that occurs exactly once. Example: a(5) = 4 because in the partitions of 4, namely [1,1,1,1], [1,1,2'], [2,2], [1',3'], [4'] we have 4 singletons (marked by '). - _Emeric Deutsch_, Sep 12 2016
%C a(n) is also the number of non-isomorphic vertex-transitive cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_{n-1}. See Table 1 in the Hoang/Mütze reference in the Links section. - _Torsten Muetze_, Nov 28 2019
%C Assuming a partition is in weakly decreasing order, a(n) is also the number of times -1 occurs in the differences of the partitions of n+1. - _George Beck_, Mar 28 2023
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 184.
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A024786/b024786.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%H David Benson, Radha Kessar, and Markus Linckelmann, <a href="https://arxiv.org/abs/2204.09970">Hochschild cohomology of symmetric groups in low degrees</a>, arXiv:2204.09970 [math.GR], 2022.
%H Manosij Ghosh Dastidar and Sourav Sen Gupta, <a href="http://arxiv.org/abs/1111.0094">Generalization of a few results in Integer Partitions</a>, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
%H Emeric Deutsch et al., <a href="https://www.mat.uniroma2.it/~tauraso/AMM/AMM11237.pdf">Problem 11237</a>, Amer. Math. Monthly, 115 (No. 7, 2008), 666-667. [From _Emeric Deutsch_, Aug 13 2008]
%H Hung Phuc Hoang and Torsten Mütze, <a href="https://arxiv.org/abs/1911.12078">Combinatorial generation via permutation languages. II. Lattice congruences</a>, arXiv:1911.12078 [math.CO], 2019.
%H Joseph Vandehey, <a href="https://math.colgate.edu/~integers/a18Proc23/a18Proc23.pdf">Digital problems in the theory of partitions</a>, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.
%F a(n) = Sum_{k=1..floor(n/2)} A000041(n-2k). - _Christian G. Bower_, Jun 22 2000
%F a(n) = Sum_{k<n, k = n (mod 2)} P(k), P(k) = number of partitions of k as in A000041, P(0) = 1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
%F G.f.: x^2/((1-x)*(1-x^2)^2))*Product_{j>=3} 1/(1-x^j) from Riordan reference second term, last eq.
%F a(n) = A006128(n-1) - A194452(n-1). - _Omar E. Pol_, Nov 20 2011
%F a(n) = A181187(n,2) - A181187(n,3). - _Omar E. Pol_, Oct 25 2012
%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2) * Pi * sqrt(n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 433*Pi^2/6912)/n). - _Vaclav Kotesovec_, Mar 07 2016, extended Nov 05 2016
%F a(n) = Sum_{k} k * A116595(n-1,k). - _Emeric Deutsch_, Sep 12 2016
%F G.f.: x^2/((1 - x)*(1 - x^2)) * Sum_{n >= 0} x^(2*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A004526 (partitions into 2 parts, or, modulo offset differences, partitions into parts <= 2) and A002865 (partitions into parts >= 2). - _Peter Bala_, Jan 17 2021
%e From _Omar E. Pol_, Oct 25 2012: (Start)
%e For n = 7 we have:
%e --------------------------------------
%e . Number
%e Partitions of 7 of 2's
%e --------------------------------------
%e 7 .............................. 0
%e 4 + 3 .......................... 0
%e 5 + 2 .......................... 1
%e 3 + 2 + 2 ...................... 2
%e 6 + 1 .......................... 0
%e 3 + 3 + 1 ...................... 0
%e 4 + 2 + 1 ...................... 1
%e 2 + 2 + 2 + 1 .................. 3
%e 5 + 1 + 1 ...................... 0
%e 3 + 2 + 1 + 1 .................. 1
%e 4 + 1 + 1 + 1 .................. 0
%e 2 + 2 + 1 + 1 + 1 .............. 2
%e 3 + 1 + 1 + 1 + 1 .............. 0
%e 2 + 1 + 1 + 1 + 1 + 1 .......... 1
%e 1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
%e ------------------------------------
%e . 24 - 13 = 11
%e .
%e The difference between the sum of the second column and the sum of the third column of the set of partitions of 7 is 24 - 13 = 11 and equals the number of 2's in all partitions of 7, so a(7) = 11.
%e (End)
%p b:= proc(n, i) option remember; local f, g;
%p if n=0 or i=1 then [1, 0]
%p else f:= b(n, i-1); g:= `if`(i>n, [0$2], b(n-i, i));
%p [f[1]+g[1], f[2]+g[2]+`if`(i=2, g[1], 0)]
%p fi
%p end:
%p a:= n-> b(n, n)[2]:
%p seq(a(n), n=1..50); # _Alois P. Heinz_, May 18 2012
%t Table[ Count[ Flatten[ IntegerPartitions[n]], 2], {n, 1, 50} ]
%t (* Second program: *)
%t b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, 0}, f = b[n, i - 1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i == 2, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Sep 22 2015, after _Alois P. Heinz_ *)
%t Join[{0}, (1/((1 - x^2) QPochhammer[x]) + O[x]^50)[[3]]] (* _Vladimir Reshetnikov_, Nov 22 2016 *)
%t Table[Sum[(1 + (-1)^k)/2 * PartitionsP[n-k], {k, 2, n}], {n, 1, 50}] (* _Vaclav Kotesovec_, Aug 27 2017 *)
%o (Python)
%o from sympy import npartitions
%o def A024786(n): return sum(npartitions(n-(k<<1)) for k in range(1,(n>>1)+1)) # _Chai Wah Wu_, Oct 25 2023
%Y Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A173238.
%Y Column 2 of A060244.
%Y First differences of A000097.
%K nonn
%O 1,4
%A _Clark Kimberling_
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