A092920 Number of strongly monotone partitions of [n].

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

Offset: 0

Author

Ralf Stephan, Apr 17 2004

Keywords

• nonn

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.

Crossrefs

Cf. A001006, A006789.

Similar recurrences: A284005, A329369, A341392.

Programs

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 *)

Formula

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.

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

Extensions

More terms from Emeric Deutsch, Apr 13 2005