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 A092920 Number of strongly monotone partitions of [n]. 4
 1, 1, 2, 4, 9, 22, 58, 164, 496, 1601, 5502, 20075, 77531, 315947, 1354279, 6087421, 28611385, 140239297, 715116827, 3785445032, 20760746393, 117759236340, 689745339984, 4165874930885, 25911148634728, 165775085602106, 1089773992530717, 7353740136527305 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A partition is strongly monotone if its blocks can be written in increasing order of their least element and increasing order of their greatest element, simultaneously. a(n) is the number of strongly nonoverlapping partitions of [n] where "strongly nonoverlapping" means nonoverlapping (see A006789 for definition) and, in addition, no singleton block is a subset of the span (interval from minimum to maximum) of another block. For example, 13-24 is nonnesting and 14-23 is strongly nonoverlapping but neither has the other property. The Motzkin number M_n (A001006) counts strongly noncrossing partitions of [n]. - David Callan, Sep 20 2007 Strongly monotone partitions can also be described as partitions in which no block is contained in the span of another, where span denotes the interval from smallest to largest entries. For example, 134/25/6 is strongly monotone but 135/24/6 is not because the block 24 is contained in the interval [1,5]. - David Callan, Aug 27 2014 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 A. Claesson and T. Mansour, Enumerating permutations avoiding a pair of Babson-Steingrimsson patterns, arXiv:math/0107044 [math.CO], 2001-2010. FORMULA G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-x-x^2/(1-x-x^2/(1-2x-x^2/(1-3x-x^2/...))) = 1/(1-x-x^2*B(x)) where B(x) is g.f. for the Bessel numbers A006789. a(n) = leftmost column terms of M^n*V, where M = an infinite tridiagonal matrix with all 1's in the super and subdiagonals and (1,1,2,3,4,5,...) as the main diagonal; and the rest zeros. V = vector [1,0,0,0,...]. - Gary W. Adamson, Jun 16 2011 G.f.: 1/Q(0) where Q(k) = 1-x*(k+2)+x/(1+x/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013 Conjecture: a(n) = Sum_{k=0..2^(n-1) - 1} b(k) for n > 0 with a(0) = 1 where b(2^m*(2n+1)) = Sum_{k=0..[m > 0]*(m-1)} binomial(m-1, k)*b(2^k*n) for m >= 0, n >= 0 with b(0) = 1. - Mikhail Kurkov, Apr 24 2023 MAPLE G:=1/(1-x-x^2/(1-x-x^2/(1-2*x-x^2/(1-3*x-x^2/(1-4*x-x^2/(1-5*x-x^2/(1-6*x-x^2/(1-7*x-x^2/(1-8*x-x^2/(1-9*x-x^2/(1-10*x-x^2/(1-11*x-x^2/(1-12*x-x^2/(1-13*x-x^2/(1-14*x-x^2/(1-15*x-x^2/(1-16*x-x^2/(1-17*x-x^2)))))))))))))))))): Gser:=series(G, x=0, 32): seq(coeff(Gser, x, n), n=0..28); # Emeric Deutsch, Apr 13 2005 MATHEMATICA terms = 26; f[1] = 1; f[k_ /; k>1] = -x^2; g[1] = 1-x; g[k_ /; k>1] := 1-(k-1)x; A[x_] = ContinuedFractionK[f[k], g[k], {k, 1, Ceiling[terms/2]}]; CoefficientList[A[x] + O[x]^terms, x] (* Jean-François Alcover, Aug 07 2018 *) CROSSREFS Cf. A001006, A006789. Similar recurrences: A284005, A329369, A341392. Sequence in context: A121953 A024427 A171367 * A177377 A321994 A035053 Adjacent sequences: A092917 A092918 A092919 * A092921 A092922 A092923 KEYWORD nonn AUTHOR Ralf Stephan, Apr 17 2004 EXTENSIONS More terms from Emeric Deutsch, Apr 13 2005 STATUS approved

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