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A092920 Number of strongly monotone partitions of [n].

%I A092920 #52 Apr 30 2025 09:15:08
%S A092920 1,1,2,4,9,22,58,164,496,1601,5502,20075,77531,315947,1354279,6087421,
%T A092920 28611385,140239297,715116827,3785445032,20760746393,117759236340,
%U A092920 689745339984,4165874930885,25911148634728,165775085602106,1089773992530717,7353740136527305
%N A092920 Number of strongly monotone partitions of [n].
%C A092920 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.
%C A092920 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
%C A092920 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
%H A092920 Alois P. Heinz, <a href="/A092920/b092920.txt">Table of n, a(n) for n = 0..500</a>
%H A092920 A. Claesson and T. Mansour, <a href="https://arxiv.org/abs/math/0107044">Enumerating permutations avoiding a pair of Babson-Steingrimsson patterns</a>, arXiv:math/0107044 [math.CO], 2001-2010.
%F A092920 G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-x-x^2/(1-x-x^2/(1-2*x-x^2/(1-3*x-x^2/...)))) = 1/(1-x-x^2*B(x)) where B(x) is g.f. for the Bessel numbers A006789.
%F A092920 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
%F A092920 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
%F A092920 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
%p A092920 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
%t A092920 terms = 26;
%t A092920 f[1] = 1; f[k_ /; k>1] = -x^2;
%t A092920 g[1] = 1-x; g[k_ /; k>1] := 1-(k-1)x;
%t A092920 A[x_] = ContinuedFractionK[f[k], g[k], {k, 1, Ceiling[terms/2]}];
%t A092920 CoefficientList[A[x] + O[x]^terms, x] (* _Jean-François Alcover_, Aug 07 2018 *)
%Y A092920 Cf. A001006, A006789.
%Y A092920 Similar recurrences: A284005, A329369, A341392.
%K A092920 nonn
%O A092920 0,3
%A A092920 _Ralf Stephan_, Apr 17 2004
%E A092920 More terms from _Emeric Deutsch_, Apr 13 2005