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%I M0309 N0113 #312 Feb 08 2024 11:32:35
%S 1,0,1,1,2,2,4,4,7,8,12,14,21,24,34,41,55,66,88,105,137,165,210,253,
%T 320,383,478,574,708,847,1039,1238,1507,1794,2167,2573,3094,3660,4378,
%U 5170,6153,7245,8591,10087,11914,13959,16424,19196,22519,26252,30701
%N Number of partitions of n that do not contain 1 as a part.
%C Also the number of partitions of n-1, n >= 2, such that the least part occurs exactly once. See A096373, A097091, A097092, A097093. - _Robert G. Wilson v_, Jul 24 2004 [Corrected by _Wolfdieter Lang_, Feb 18 2009]
%C Number of partitions of n+1 where the number of parts is itself a part. Take a partition of n (with k parts) which does not contain 1, remove 1 from each part and add a new part of size k+1. - _Franklin T. Adams-Watters_, May 01 2006
%C Number of partitions where the largest part occurs at least twice. - _Joerg Arndt_, Apr 17 2011
%C Row sums of triangle A147768. - _Gary W. Adamson_, Nov 11 2008
%C From Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)
%C a(n) is the number of sets of n disjoint pairs of 2n things, called a pairing, disjoint with a given pairing (A053871), that are unique under permutations preserving the given pairing.
%C Can be seen immediately from a graphical representation which must decompose into even numbered cycles of 4 or more things, as connected by pairs alternating between the pairings. Each thing is in a single cycle, so this is a partition of 2n into even parts greater than 2, equivalent to a partition of n into parts greater than 1. (End)
%C Convolution product (1, 1, 2, 2, 4, 4, ...) * (1, 2, 3, ...) = A058682 starting (1, 3, 7, 13, 23, 37, ...); with row sums of triangle A171239 = A058682. - _Gary W. Adamson_, Dec 05 2009
%C Also the number of 2-regular multigraphs with loops forbidden. - _Jason Kimberley_, Jan 05 2011
%C Number of appearances of the multiplicity n, n-1, ..., n-k in all partitions of n, for k < n/2. (Only populated by multiplicities of large numbers of 1's.) - William Keith, Nov 20 2011
%C Also the number of equivalence classes of n X n binary matrices with exactly 2 1's in each row and column, up to permutations of rows and columns (cf. A133687). - _N. J. A. Sloane_, Sep 16 2013
%C The q-Catalan numbers ((1-q)/(1-q^(n+1)))[2n,n]_q, where [2n,n]_q are the central q-binomial coefficients, match this sequence in their initial segment of length n. - _William J. Keith_, Nov 14 2013
%C Starting at a(2) this sequence gives the number of vertices on a nim tree created in the game of edge removal for a path P_{n} where n is the number of vertices on the path. This is the number of non-isomorphic graphs that can result from the path when the game of edge removal is played. - _Lyndsey Wong_, Jul 09 2016
%C The number of different ways to climb a staircase taking at least two stairs at a time. - _Mohammad K. Azarian_, Nov 20 2016
%C Let 1,0,1,1,1,... (offset 0) count unlabeled, connected, loopless 1-regular digraphs. This here is the Euler transform of that sequence, counting unlabeled loopless 1-regular digraphs. A145574 is the associated multiset transformation. A000166 are the labeled loopless 1-regular digraphs. - _R. J. Mathar_, Mar 25 2019
%C For n > 1, also the number of partitions with no part greater than the number of ones. - _George Beck_, May 09 2019 [See A187219 which is the correct sequence for this interpretation for n >= 1. - _Spencer Miller_, Jan 30 2023]
%C From _Gus Wiseman_, May 19 2019: (Start)
%C Conjecture: Also the number of integer partitions of n - 1 that have a consecutive subsequence summing to each positive integer from 1 to n - 1. For example, (32211) is such a partition because we have consecutive subsequences:
%C 1: (1)
%C 2: (2)
%C 3: (3) or (21)
%C 4: (22) or (211)
%C 5: (32) or (221)
%C 6: (2211)
%C 7: (322)
%C 8: (3221)
%C 9: (32211)
%C (End)
%C There is a sufficient and necessary condition to characterize the partitions defined by Gus Wiseman. It is that the largest part must be less than or equal to the number of ones plus one. Hence, the number of partitions of n with no part greater than the number of ones is the same as the number of partitions of n-1 that have a consecutive subsequence summing to each integer from 1 to n-1. Gus Wiseman's conjecture can be proved bijectively. - _Andrew Yezhou Wang_, Dec 14 2019
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n).
%D H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 334.
%H Andrew van den Hoeven, <a href="/A002865/b002865.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A. P. Akande et al., <a href="https://arxiv.org/abs/2112.03264">Computational study of non-unitary partitions</a>, arXiv:2112.03264 [math.CO], 2021.
%H Colin Albert, Olivia Beckwith, Irfan Demetoglu, Robert Dicks, John H. Smith, and Jasmine Wang, <a href="https://arxiv.org/abs/2203.08987">Integer partitions with large Dyson rank</a>, arXiv:2203.08987 [math.NT], 2022.
%H Kevin Beanland and Hung Viet Chu, <a href="https://arxiv.org/abs/2311.01926">On Schreier-type Sets, Partitions, and Compositions</a>, arXiv:2311.01926 [math.CO], 2023.
%H G. Dahl and T. A. Haufmann, <a href="https://www.researchgate.net/publication/303684695_Zero-one_completely_positive_matrices_and_the_AR_S_classes">Zero-one completely positive matrices and the A(R,S) classes</a>, Preprint, 2016.
%H R. P. Gallant, G. Gunther, B. L. Hartnell, and D. F. Rall, <a href="https://www.researchgate.net/publication/268659207">A game of edge removal on graphs</a>, JCMCC, 57 (2006), 75 - 82.
%H Edray Herber Goins and Talitha M. Washington, <a href="https://arxiv.org/abs/0909.5459">On the generalized climbing stairs problem</a>, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
%H H. Gropp, <a href="http://dx.doi.org/10.1016/0012-365X(94)00372-P">On tactical configurations, regular bipartite graphs and (v,k,even)-designs</a>, Discr. Math., 155 (1996), 81-98.
%H R. K. Guy and N. J. A. Sloane, <a href="/A005180/a005180.pdf">Correspondence</a>, 1988.
%H Cristiano Husu, <a href="https://arxiv.org/abs/1804.09883">The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2</a>, arXiv:1804.09883 [math.NT], 2018.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=100">Encyclopedia of Combinatorial Structures 100</a>
%H Wenwei Li, <a href="http://arxiv.org/abs/1612.05526">Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics</a>, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018.
%H Wenwei Li, <a href="https://arxiv.org/abs/1612.08186">On the Number of Conjugate Classes of Derangements</a>, arXiv:1612.08186 [math.CO], 2016.
%H J. L. Nicolas and A. Sárközy, <a href="http://www.numdam.org/item?id=JTNB_2000__12_1_227_0">On partitions without small parts</a>, Journal de théorie des nombres de Bordeaux, 12 no. 1 (2000), p. 227-254.
%H R. A. Proctor, <a href="https://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.
%H H. P. Robinson, <a href="/A002065/a002065.pdf">Letter to N. J. A. Sloane, Jul 12 1971</a>
%H H. P. Robinson, <a href="/A002865/a002865.pdf">Letter to N. J. A. Sloane, Dec 10 1973</a>
%H H. P. Robinson, <a href="/A003105/a003105.pdf">Letter to N. J. A. Sloane, Jan 4 1974</a>.
%H Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, <a href="https://arxiv.org/abs/2103.11018">Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares</a>, arXiv:2103.11018 [cs.DM], 2021.
%H Miloslav Znojil, <a href="https://arxiv.org/abs/2010.15014">Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities</a>, arXiv:2010.15014 [quant-ph], 2020.
%H Miloslav Znojil, <a href="https://arxiv.org/abs/2102.12272">Quantum phase transitions mediated by clustered non-Hermitian degeneracies</a>, arXiv:2102.12272 [quant-ph], 2021.
%H Miloslav Znojil, <a href="https://arxiv.org/abs/2108.07110">Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks</a>, arXiv:2108.07110 [quant-ph], 2021.
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F G.f.: Product_{m>1} 1/(1-x^m).
%F a(0)=1, a(n) = p(n) - p(n-1), n >= 1, with the partition numbers p(n) := A000041(n).
%F a(n) = A085811(n+3). - _James A. Sellers_, Dec 06 2005 [Corrected by _Gionata Neri_, Jun 14 2015]
%F a(n) = A116449(n) + A116450(n). - _Reinhard Zumkeller_, Feb 16 2006
%F a(n) = Sum_{k=2..floor((n+2)/2)} A008284(n-k+1,k-1) for n > 0. - _Reinhard Zumkeller_, Nov 04 2007
%F G.f.: 1 + Sum_{n>=2} x^n / Product_{k>=n} (1 - x^k). - _Joerg Arndt_, Apr 13 2011
%F G.f.: Sum_{n>=0} x^(2*n) / Product_{k=1..n} (1 - x^k). - _Joerg Arndt_, Apr 17 2011
%F a(n) = A090824(n,1) for n > 0. - _Reinhard Zumkeller_, Oct 10 2012
%F a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - _Vaclav Kotesovec_, Feb 26 2015, extended Nov 04 2016
%F G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - _Ilya Gutkovskiy_, Aug 21 2018
%F a(0) = 1, a(n) = A232697(n) - 1. - _George Beck_, May 09 2019
%F From _Peter Bala_, Feb 19 2021: (Start)
%F G.f.: A(q) = Sum_{n >= 0} q^(n^2)/( (1 - q)*Product_{k = 2..n} (1 - q^k)^2 ).
%F More generally, A(q) = Sum_{n >= 0} q^(n*(n+r))/( (1 - q) * Product_{k = 2..n} (1 - q^k)^2 * Product_{i = 1..r} (1 - q^(n+i)) ) for r = 0,1,2,.... (End)
%e a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.
%e G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 7*x^8 + 8*x^9 + ...
%e From _Gus Wiseman_, May 19 2019: (Start)
%e The a(2) = 1 through a(9) = 8 partitions not containing 1 are the following. The Heinz numbers of these partitions are given by A005408.
%e (2) (3) (4) (5) (6) (7) (8) (9)
%e (22) (32) (33) (43) (44) (54)
%e (42) (52) (53) (63)
%e (222) (322) (62) (72)
%e (332) (333)
%e (422) (432)
%e (2222) (522)
%e (3222)
%e The a(2) = 1 through a(9) = 8 partitions of n - 1 whose least part appears exactly once are the following. The Heinz numbers of these partitions are given by A247180.
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (21) (31) (32) (42) (43) (53)
%e (41) (51) (52) (62)
%e (221) (321) (61) (71)
%e (331) (332)
%e (421) (431)
%e (2221) (521)
%e (3221)
%e The a(2) = 1 through a(9) = 8 partitions of n + 1 where the number of parts is itself a part are the following. The Heinz numbers of these partitions are given by A325761.
%e (21) (22) (32) (42) (52) (62) (72) (82)
%e (311) (321) (322) (332) (333) (433)
%e (331) (431) (432) (532)
%e (4111) (4211) (531) (631)
%e (4221) (4222)
%e (4311) (4321)
%e (51111) (4411)
%e (52111)
%e The a(2) = 1 through a(8) = 7 partitions of n whose greatest part appears at least twice are the following. The Heinz numbers of these partitions are given by A070003.
%e (11) (111) (22) (221) (33) (331) (44)
%e (1111) (11111) (222) (2221) (332)
%e (2211) (22111) (2222)
%e (111111) (1111111) (3311)
%e (22211)
%e (221111)
%e (11111111)
%e Non-isomorphic representatives of the a(2) = 1 through a(6) = 4 2-regular multigraphs with n edges and n vertices are the following.
%e {12,12} {12,13,23} {12,12,34,34} {12,12,34,35,45} {12,12,34,34,56,56}
%e {12,13,24,34} {12,13,24,35,45} {12,12,34,35,46,56}
%e {12,13,23,45,46,56}
%e {12,13,24,35,46,56}
%e The a(2) = 1 through a(9) = 8 partitions of n with no part greater than the number of ones are the following. The Heinz numbers of these partitions are given by A325762.
%e (11) (111) (211) (2111) (2211) (22111) (22211) (33111)
%e (1111) (11111) (3111) (31111) (32111) (222111)
%e (21111) (211111) (41111) (321111)
%e (111111) (1111111) (221111) (411111)
%e (311111) (2211111)
%e (2111111) (3111111)
%e (11111111) (21111111)
%e (111111111)
%e (End)
%p with(combstruct): ZL1:=[S, {S=Set(Cycle(Z,card>1))}, unlabeled]: seq(count(ZL1,size=n), n=0..50); # _Zerinvary Lajos_, Sep 24 2007
%p G:= {P=Set (Set (Atom, card>1))}: combstruct[gfsolve](G, unlabeled, x): seq (combstruct[count] ([P, G, unlabeled], size=i), i=0..50); # _Zerinvary Lajos_, Dec 16 2007
%p with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, unlabeled]; end: A:=a(2):seq(count(A, size=n), n=0..50); # _Zerinvary Lajos_, Jun 11 2008
%p # alternative Maple program:
%p A002865:= proc(n) option remember; `if`(n=0, 1, add(
%p (numtheory[sigma](j)-1)*A002865(n-j), j=1..n)/n)
%p end:
%p seq(A002865(n), n=0..60); # _Alois P. Heinz_, Sep 17 2017
%t Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (* _Robert G. Wilson v_, Jul 24 2004 *)
%t f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 2], {n, 50}] (* _Robert G. Wilson v_ *)
%t Table[SeriesCoefficient[Exp[Sum[x^(2*k)/(k*(1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 50}] (* _Vaclav Kotesovec_, Aug 18 2018 *)
%t CoefficientList[Series[1/QPochhammer[x^2, x], {x,0,50}], x] (* _G. C. Greubel_, Nov 03 2019 *)
%t Table[Count[IntegerPartitions[n],_?(FreeQ[#,1]&)],{n,0,50}] (* _Harvey P. Dale_, Feb 12 2023 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - x) / eta(x + x * O(x^n)), n))};
%o (PARI) a(n)=if(n,numbpart(n)-numbpart(n-1),1) \\ _Charles R Greathouse IV_, Nov 26 2012
%o (Magma) A41 := func<n|n ge 0 select NumberOfPartitions(n) else 0>; [A41(n)-A41(n-1):n in [0..50]]; // _Jason Kimberley_, Jan 05 2011
%o (GAP) Concatenation([1],List([1..41],n->NrPartitions(n)-NrPartitions(n-1))); # _Muniru A Asiru_, Aug 20 2018
%o (Sage)
%o def A002865_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/product((1-x^(m+2)) for m in (0..60)) ).list()
%o A002865_list(50) # _G. C. Greubel_, Nov 03 2019
%o (Python)
%o from sympy import npartitions
%o def A002865(n): return npartitions(n)-npartitions(n-1) if n else 1 # _Chai Wah Wu_, Mar 30 2023
%Y First differences of partition numbers A000041. Cf. A053445, A072380, A081094, A081095, A232697.
%Y Pairwise sums seem to be in A027336.
%Y Essentially the same as A085811.
%Y Cf. A025147, A147768, A058682, A171239.
%Y A column of A090824 and of A133687 and of A292508 and of A292622. Cf. A229161.
%Y 2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), this sequence (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges). - _Jason Kimberley_, Jan 05 2011
%Y See also A098743 (parts that do not divide n).
%Y Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy: A274161. - _Lyndsey Wong_, Jul 09 2016
%Y Cf. A002033, A070003, A103295, A247180, A325676, A325684, A325761, A325762, A325763.
%Y Row suns of characteristic array A145573.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_
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