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 A002865 Number of partitions of n that do not contain 1 as a part. (Formerly M0309 N0113) 251
 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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] 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 Number of partitions where the largest part occurs at least twice. - Joerg Arndt, Apr 17 2011 Row sums of triangle A147768. - Gary W. Adamson, Nov 11 2008 From Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start) 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. 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) 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 Also the number of 2-regular multigraphs with loops forbidden. - Jason Kimberley, Jan 05 2011 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 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 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 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 The number of different ways to climb a staircase taking at least two stairs at a time. - Mohammad K. Azarian, Nov 20 2016 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 Also the number of partitions with no part greater than the number of ones. - George Beck, May 09 2019 From Gus Wiseman, May 19 2019: (Start) 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:   1: (1)   2: (2)   3: (3) or (21)   4: (22) or (211)   5: (32) or (221)   6: (2211)   7: (322)   8: (3221)   9: (32211) (End) 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 REFERENCES 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. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n). H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 334. LINKS T. D. Noe and Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe) M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. A. P. Akande et al., Computational study of non-unitary partitions, arXiv:2112.03264 [math.CO], 2021. Colin Albert, Olivia Beckwith, Irfan Demetoglu, Robert Dicks, John H. Smith, and Jasmine Wang, Integer partitions with large Dyson rank, arXiv:2203.08987 [math.NT], 2022. G. Dahl and T. A. Haufmann, Zero-one completely positive matrices and the A(R,S) classes, Preprint, 2016. R. P. Gallant, G. Gunther, B. L. Hartnell, and D. F. Rall, A game of edge removal on graphs, JCMCC, 57 (2006), 75 - 82. Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin.  117  (2014), 183-190.  MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009. H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98. R. K. Guy and N. J. A. Sloane, Correspondence, 1988. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 100 Wenwei Li, Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018. Wenwei Li, On the Number of Conjugate Classes of Derangements, arXiv:1612.08186 [math.CO], 2016. J. L. Nicolas and A. Sárközy, On partitions without small parts, Journal de théorie des nombres de Bordeaux, 12 no. 1 (2000), p. 227-254. R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007. H. P. Robinson, Letter to N. J. A. Sloane, Jul 12 1971 H. P. Robinson, Letter to N. J. A. Sloane, Dec 10 1973 H. P. Robinson, Letter to N. J. A. Sloane, Jan 4 1974. Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares, arXiv:2103.11018 [cs.DM], 2021. Miloslav Znojil, Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities, arXiv:2010.15014 [quant-ph], 2020. Miloslav Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, arXiv:2102.12272 [quant-ph], 2021. Miloslav Znojil, Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks, arXiv:2108.07110 [quant-ph], 2021. FORMULA G.f.: Product_{m>1} 1/(1-x^m). a(0)=1, a(n) = p(n)-p(n-1), n>=1, with the partition numbers p(n) := A000041(n). a(n) = A085811(n+3). - James A. Sellers, Dec 06 2005 [Corrected by Gionata Neri, Jun 14 2015] a(n) = A116449(n) + A116450(n). - Reinhard Zumkeller, Feb 16 2006 a(n) = Sum(A008284(n-k+1,k-1): 10. - Reinhard Zumkeller, Nov 04 2007 G.f.: 1 + sum(n>=2, x^n / prod(k>=n, 1-x^k)). - Joerg Arndt, Apr 13 2011 G.f.: sum(n>=0, x^(2*n) / prod(k=1..n, 1-x^k ) ). - Joerg Arndt, Apr 17 2011 a(n) = A090824(n,1) for n > 0. - Reinhard Zumkeller, Oct 10 2012 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 G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018 a(0) = 1, a(n) = A232697(n) - 1. - George Beck, May 09 2019 From Peter Bala, Feb 19 2021: (Start) G.f.: A(q) = Sum_{n >= 0} q^(n^2)/( (1 - q)*Product_{k = 2..n} (1 - q^k)^2 ). 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) EXAMPLE a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2. 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 + ... From Gus Wiseman, May 19 2019: (Start) The a(2) = 1 though a(9) = 8 partitions not containing 1 are the following. The Heinz numbers of these partitions are given by A005408.   (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)             (22)  (32)  (33)   (43)   (44)    (54)                         (42)   (52)   (53)    (63)                         (222)  (322)  (62)    (72)                                       (332)   (333)                                       (422)   (432)                                       (2222)  (522)                                               (3222) The a(2) = 1 though 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.   (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)             (21)  (31)  (32)   (42)   (43)    (53)                         (41)   (51)   (52)    (62)                         (221)  (321)  (61)    (71)                                       (331)   (332)                                       (421)   (431)                                       (2221)  (521)                                               (3221) The a(2) = 1 though 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.   (21)  (22)  (32)   (42)   (52)    (62)    (72)     (82)               (311)  (321)  (322)   (332)   (333)    (433)                             (331)   (431)   (432)    (532)                             (4111)  (4211)  (531)    (631)                                             (4221)   (4222)                                             (4311)   (4321)                                             (51111)  (4411)                                                      (52111) The a(2) = 1 though 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.   (11)  (111)  (22)    (221)    (33)      (331)      (44)                (1111)  (11111)  (222)     (2221)     (332)                                 (2211)    (22111)    (2222)                                 (111111)  (1111111)  (3311)                                                      (22211)                                                      (221111)                                                      (11111111) Non-isomorphic representatives of the a(2) = 1 through a(6) = 4 2-regular multigraphs with n edges and n vertices are the following.   {12,12}  {12,13,23}  {12,12,34,34}  {12,12,34,35,45}  {12,12,34,34,56,56}                        {12,13,24,34}  {12,13,24,35,45}  {12,12,34,35,46,56}                                                         {12,13,23,45,46,56}                                                         {12,13,24,35,46,56} The a(2) = 1 though 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.   (11)  (111)  (211)   (2111)   (2211)    (22111)    (22211)     (33111)                (1111)  (11111)  (3111)    (31111)    (32111)     (222111)                                 (21111)   (211111)   (41111)     (321111)                                 (111111)  (1111111)  (221111)    (411111)                                                      (311111)    (2211111)                                                      (2111111)   (3111111)                                                      (11111111)  (21111111)                                                                  (111111111) (End) MAPLE with(combstruct): ZL1:=[S, {S=Set(Cycle(Z, card>1))}, unlabeled]: seq(count(ZL1, size=n), n=0..50);  # Zerinvary Lajos, Sep 24 2007 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 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 # alternative Maple program: a:= proc(n) option remember; `if`(n=0, 1, add(       (numtheory[sigma](j)-1)*a(n-j), j=1..n)/n)     end: seq(a(n), n=0..60);  # Alois P. Heinz, Sep 17 2017 MATHEMATICA Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (* Robert G. Wilson v, Jul 24 2004 *) 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 *) 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 *) CoefficientList[Series[1/QPochhammer[x^2, x], {x, 0, 50}], x] (* G. C. Greubel, Nov 03 2019 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - x) / eta(x + x * O(x^n)), n))}; (PARI) a(n)=if(n, numbpart(n)-numbpart(n-1), 1) \\ Charles R Greathouse IV, Nov 26 2012 (Magma) A41 := func; [A41(n)-A41(n-1):n in [0..50]]; // Jason Kimberley, Jan 05 2011 (GAP) Concatenation(, List([1..41], n->NrPartitions(n)-NrPartitions(n-1))); # Muniru A Asiru, Aug 20 2018 (Sage) def A002865_list(prec):     P. = PowerSeriesRing(ZZ, prec)     return P( 1/product((1-x^(m+2)) for m in (0..60)) ).list() A002865_list(50) # G. C. Greubel, Nov 03 2019 CROSSREFS First differences of partition numbers A000041. Cf. A053445, A072380, A081094, A081095, A232697. Pairwise sums seem to be in A027336. Essentially the same as A085811. Cf. A025147, A147768, A058682, A171239. A column of A090824 and of A133687 and of A292508 and of A292622. Cf. A229161. 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 See also A098743 (parts that do not divide n). 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 Cf. A002033, A070003, A103295, A247180, A325676, A325684, A325761, A325762, A325763. Sequence in context: A035989 A240019 A036000 * A085811 A187219 A317785 Adjacent sequences:  A002862 A002863 A002864 * A002866 A002867 A002868 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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