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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
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
Similar recurrences: A284005, A329369, A341392.
Sequence in context: A121953 A024427 A171367 * A177377 A321994 A035053
KEYWORD
nonn
AUTHOR
Ralf Stephan, Apr 17 2004
EXTENSIONS
More terms from Emeric Deutsch, Apr 13 2005
STATUS
approved

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Last modified May 26 05:37 EDT 2024. Contains 372807 sequences. (Running on oeis4.)