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 A000296 Set partitions without singletons: number of partitions of an n-set into blocks of size > 1. Also number of cyclically spaced (or feasible) partitions. (Formerly M3423 N1387) 114
 1, 0, 1, 1, 4, 11, 41, 162, 715, 3425, 17722, 98253, 580317, 3633280, 24011157, 166888165, 1216070380, 9264071767, 73600798037, 608476008122, 5224266196935, 46499892038437, 428369924118314, 4078345814329009, 40073660040755337, 405885209254049952, 4232705122975949401 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS a(n+2) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = A000110(k) for k = 0, 1, ..., n. - Michael Somos, Oct 07 2003 Number of complete rhyming schemes. Whereas the Bell number B(n) (A000110(n)) is the number of terms in the polynomial that expresses the n-th moment of a probability distribution as a function of the first n cumulants, these numbers give the number of terms in the corresponding expansion of the _central_ moment as a function of the first n cumulants. - Michael Hardy (hardy(AT)math.umn.edu), Jan 26 2005 a(n) is the number of permutations on [n] for which the left-to-right maxima coincide with the descents (entries followed by a smaller number). For example, a(4) counts 2143, 3142, 3241, 4123. - David Callan, Jul 20 2005 From Gus Wiseman, Feb 10 2019: (Start) Also the number of stable partitions of an n-cycle, where a stable partition of a graph is a set partition of the vertex set such that no edge has both ends in the same block. A bijective proof is given in David Callan's article. For example, the a(5) = 11 stable partitions are:   {{1},{2},{3},{4},{5}}   {{1},{2},{3,5},{4}}   {{1},{2,4},{3},{5}}   {{1},{2,5},{3},{4}}   {{1,3},{2},{4},{5}}   {{1,4},{2},{3},{5}}   {{1},{2,4},{3,5}}   {{1,3},{2,4},{5}}   {{1,3},{2,5},{4}}   {{1,4},{2},{3,5}}   {{1,4},{2,5},{3}} (End) Also number of partitions of {1, 2, ..., n-1} with singletons. E.g., a(4) = 4: {1|2|3, 12|3, 13|2, 1|23}. Also number of cyclical adjacencies partitions of {1, 2, ..., n-1}. E.g., a(4) = 4: {12|3, 13|2, 1|23, 123}. The two partitions can be mapped by a Kreweras bijection. - Yuchun Ji, Feb 22 2021 REFERENCES Martin Gardner in Sci. Amer. May 1977. D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 436). G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 228. Reidenbach, Daniel, and Johannes C. Schneider. "Morphically primitive words." (2009). See Table 1. Available from https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/4561/1/Reidenbach_Schneider_TCS_Morphically_primitive_words_Final_version.pdf [Do not delete this reference, because I do not know if the similar link below (which does not seem to work) refers to an identical version of the article. - N. J. A. Sloane, Jul 14 2018] J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979. J. Shallit, A Triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71. 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). LINKS T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..575 (first 101 terms from T. D. Noe) Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order" E. Bach, Random bisection and evolutionary walks, J. Applied Probability, v. 38, pp. 582-596, 2001. M. Bauer and O. Golinelli, Random incidence matrices: Moments of the spectral density, J. Stat. Phys. 103, 301-307 (2001) and ArXiv: cond-mat/0007127. See Sect. 3.2. H. D. Becker, Solution to problem E 461, American Math Monthly 48 (1941), 701-702. F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112. F. R. Bernhart, Fundamental chromatic numbers, Unpublished. (Annotated scanned copy) F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977 J. R. Britnell and M. Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv 1507.04803 [math.CO], 2015. David Callan, On conjugates for set partitions and integer compositions [math.CO]. Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, and Bruno Patrou, State complexity of catenation combined with a boolean operation: a unified approach, arXiv preprint arXiv:1505.03474 [cs.FL], 2015. Éva Czabarka, Péter L. Erdős, Virginia Johnson, Anne Kupczok and László A. Székely, Asymptotically normal distribution of some tree families relevant for phylogenetics, and of partitions without singletons, arXiv preprint arXiv:1108.6015 [math.CO], 2011. Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv preprint arXiv:1104.4323 [hep-th], 2011. E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73. Steven R. Finch, Moments of sums, April 23, 2004 [Cached copy, with permission of the author] Robert C. Griffiths, P. A. Jenkins, and S. Lessard, A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning, arXiv preprint arXiv:1604.04145 [q-bio.PE], 2016. Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 16 V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. Peter Luschny, Set partitions Gregorio Malajovich, Complexity of sparse polynomial solving: homotopy on toric varieties and the condition metric, arXiv preprint arXiv:1606.03410 [math.NA], 2016. Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67. T. Mansour and A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216. I. Mezo, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17, Article 14.1.1, 2014. E. Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013. Tilman Piesk, Table showing non-singleton partitions for n=1...6 R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007. Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29. Daniel Reidenbach and Johannes C. Schneider, Morphically primitive words, (2009). See Table 1. Daniel Reidenbach and Johannes C. Schneider, Morphically primitive words, Theoretical Computer Science, (2009), 140 (21-23), pp. 2148-2161. J. Riordan, Cached copy of paper J. Shallit, A Triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71. FORMULA E.g.f.: exp(exp(x) - 1 - x). B(n) = a(n) + a(n+1), where B = A000110 = Bell numbers [Becker]. Inverse binomial transform of Bell numbers (A000110). a(n)= Sum_{k>=-1} (k^n/(k+1)!)/exp(1). - Vladeta Jovovic and Karol A. Penson, Feb 02 2003 a(n) = Sum_{k=0..n} ((-1)^(n-k))*binomial(n, k)*Bell(k) = (-1)^n + Bell(n) - A087650(n), with Bell(n) = A000110(n). - Wolfdieter Lang, Dec 01 2003 O.g.f.: A(x) = 1/(1-0*x-1*x^2/(1-1*x-2*x^2/(1-2*x-3*x^2/(1-... -(n-1)*x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006 a(n) = Sum_{k = 0..n} {(-1)^(n-k) * Sum_{j = 0..k}((-1)^j * binomial(k,j) * (1-j)^n)/ k!} = sum over row n of A105794. - Tom Copeland, Jun 05 2006 a(n) = (-1)^n + Sum_{j=1..n} (-1)^(j-1)*B(n-j), where B(q) are the Bell numbers (A000110). - Emeric Deutsch, Oct 29 2006 Let A be the upper Hessenberg matrix of order n defined by: A[i, i-1] = -1, A[i,j] = binomial(j-1, i-1), (i <= j), and A[i, j] = 0 otherwise. Then, for n >= 2, a(n) = (-1)^(n)charpoly(A,1). - Milan Janjic, Jul 08 2010 From Sergei N. Gladkovskii, Sep 20 2012, Oct 11 2012, Dec 19 2012, Jan 15 2013, May 13 2013, Jul 20 2013, Oct 19 2013, Jan 25 2014: (Start) Continued fractions: G.f.: (2/E(0) - 1)/x where E(k) = 1 + 1/(1 + 2*x/(1 - 2*(k+1)*x/E(k+1))). G.f.: 1/U(0) where U(k) = 1 - x*k - x^2*(k+1)/U(k+1). G.f.: G(0)/(1+2*x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-x-1) - x*(2*k+1)*(2*k+3)*(2*x*k-x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k-1)/G(k+1))). G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+x-k*x)/(1-x/(x-1/G(k+1))). G.f.: 1 + x^2/(1+x)/Q(0) where Q(k) = 1-x-x/(1-x*(2*k+1)/(1-x-x/(1-x*(2*k+2)/Q(k+1)))). G.f.: 1/(x*Q(0)) where Q(k) = 1 + 1/(x + x^2/(1 - x - (k+1)/Q(k+1))). G.f.: -(1+(2*x+1)/G(0))/x where G(k) = x*k - x - 1 - (k+1)*x^2/G(k+1). G.f.: T(0) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*k)*(1-x-x*k)/T(k+1)). G.f.: (1+x*Sum_{k>=0} x^k/Product_{p=0..k}(1-p*x)/(1+x). (End) a(n) = Sum_{i=1..n-1} binomial(n-1,i)*a(n-i-1), a(0)=1. - Vladimir Kruchinin, Feb 22 2015 G.f. A(x) satisfies: A(x) = (1/(1 + x)) * (1 + x * A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, May 21 2021 a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n-1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n-1)). - Vaclav Kotesovec, Jun 28 2022 EXAMPLE For n = 4 the a(4) = card({{{1, 2}, {3, 4}}, {{1, 4}, {2, 3}}, {{1, 3}, {2, 4}}, {{1, 2, 3, 4}}}) = 4. MAPLE spec := [ B, {B=Set(Set(Z, card>1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..30)]; with(combinat): A000296 :=n->(-1)^n + add((-1)^(j-1)*bell(n-j), j=1..n): seq(A000295(n), n=0..30); # Emeric Deutsch, Oct 29 2006 f:=exp(exp(x)-1-x): fser:=series(f, x=0, 31): 1, seq(n!*coeff(fser, x^n), n=1..23); # Zerinvary Lajos, Nov 22 2006 G:={P=Set(Set(Atom, card>=2))}: combstruct[gfsolve](G, unlabeled, x): seq(combstruct[count]([P, G, labeled], size=i), i=0..23); # Zerinvary Lajos, Dec 16 2007 # [a(0), a(1), .., a(n)] A000296_list := proc(n) local A, R, i, k; if n = 0 then RETURN(1) fi; A := array(0..n-1); A := 1; R := 1; for i from 0 to n-2 do    A[i+1] := A - A[i];    A[i] := A;    for k from i by -1 to 1 do       A[k-1] := A[k-1] + A[k] od;    R := R, A[i+1]; od; R, A-A[i] end: A000296_list(100);  # Peter Luschny, Apr 09 2011 MATHEMATICA nn = 25; Range[0, nn]! CoefficientList[Series[Exp[Exp[x] - 1 - x], {x, 0, nn}], x] (* Second program: *) a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n-1, i]*a[n-i-1], {i, 1, n-1}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2016, after Vladimir Kruchinin *) spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:= Join@@Function[s, Prepend[#, s]&/@spsu[ Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}]; Table[Length[spsu[Select[Subsets[Range[n]], Select[Partition[Range[n], 2, 1, 1], Function[ed, Complement[ed, #]=={}]]=={}&], Range[n]]], {n, 8}] (* Gus Wiseman, Feb 10 2019 *) s = 1; Join[{1}, Table[s = BellB[n] - s, {n, 0, 25}]] (* Vaclav Kotesovec, Jun 20 2022 *) PROG (PARI) a(n) = if(n<2, n==0, subst( polinterpolate( Vec( serlaplace( exp( exp( x+O(x^n)/x )-1 ) ) ) ), x, n) ) (Maxima) a(n):=if n=0 then 1 else sum(binomial(n-1, i)*a(n-i-1), i, 1, n-1); /* Vladimir Kruchinin, Feb 22 2015 */ (Magma) [1, 0] cat [ n le 1 select 1 else Bell(n)-Self(n-1) : n in [1..40]]; // Vincenzo Librandi, Jun 22 2015 (Python) from itertools import accumulate, islice def A000296_gen():     yield from (1, 0)     blist, a, b = (1, ), 0, 1     while True:         blist = list(accumulate(blist, initial = (b:=blist[-1])))         yield (a := b-a) A000296_list = list(islice(A000296_gen(), 20)) # Chai Wah Wu, Jun 22 2022 CROSSREFS Cf. A000110, A005493, A006505, A057814, A057837. A diagonal of triangle in A106436. Row sums of the triangle of associated Stirling numbers of second kind A008299. - Philippe Deléham, Feb 10 2005 Row sums of the triangle of basic multinomial coefficients A178866. - Johannes W. Meijer, Jun 21 2010 Row sums of A105794. - Peter Bala, Jan 14 2015 Row sums of A261139, main diagonal of A261137. Column k=0 of A216963. Column k=0 of A124323. Cf. A000126, A001610, A066982, A169985, A240936. Sequence in context: A214188 A214239 A278989 * A032265 A320155 A260320 Adjacent sequences:  A000293 A000294 A000295 * A000297 A000298 A000299 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms, new description from Christian G. Bower, Nov 15 1999 a(23) corrected by Sean A. Irvine, Jun 22 2015 STATUS approved

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Last modified October 2 06:08 EDT 2022. Contains 357191 sequences. (Running on oeis4.)