cp's OEIS Frontend

A141618 Triangle read by rows: number of nilpotent partial transformations (of an n-element set) of height r (height(alpha) = |Im(alpha)|), 0 <= r < n.

%I A141618 #73 Jun 23 2024 21:22:05
%S A141618 1,1,2,1,9,6,1,28,72,24,1,75,500,600,120,1,186,2700,7800,5400,720,1,
%T A141618 441,12642,73500,117600,52920,5040,1,1016,54096,571536,1764000,
%U A141618 1787520,564480,40320,1,2295,217800,3916080,21019824,40007520,27941760,6531840,362880,1,5110,839700,24555600,214326000
%N A141618 Triangle read by rows: number of nilpotent partial transformations (of an n-element set) of height r (height(alpha) = |Im(alpha)|), 0 <= r < n.
%C A141618 The sum of each row of the sequence (as a triangular array) is A000272. Second left-downward diagonal is A058877.
%C A141618 From _Tom Copeland_, Oct 26 2014: (Start)
%C A141618 With T(x,t) the e.g.f. for A055302 for the number of labeled rooted trees with n nodes and k leaves, the mirror of the row polynomials of this array are given by e^T(x,t) = exp[ t * x + (2t) * x^2/2! + (6t + 3t^2) * x^3/3! + ...] = 1 + t * x + (2t + t^2) * x^2/2! + (6t + 9t^2 + t^3) * x^3/3! + ... = 1 + Nr(x,t).
%C A141618 Equivalently, e^x-1 = Nr[Tinv(x,t),t] = t * N[t*Tinv(x,t),1/t], where N(x,t) is the e.g.f. of this array and Tinv(x,t) is the comp. inverse in x of T(x,t). Note that Nr(x,t) = t * N(x*t,1/t), and N(x,t) = t * Nr(x*t,1/t). Also, log[1 + Nr(x,t)]= x * [t + Nr(x,t)] = T(x,t).
%C A141618 E.g.f. is N(x,t)= t * {exp[T(x*t,1/t)] - 1}, and log[1 + N(x,t)/t] =  T(x*t,1/t) =  x + (2t) * x^2/2! +  (3t + 6t^2) * x^3/3! + (4t + 36t^2 + 24t^3) * x^4/4! + ... = x + (t) * x^2 + (t + 2t^2) * x^3/2! + (t + 9t^2 + 6t^3) * x^4/3! +  ... is the comp. inverse in x of x / [1 + t * (e^x - 1)].
%C A141618 The exp/log transforms (A036040/A127671) generally give associations between enumerations of sets of connected graphs/objects (in this case, trees) and sets of disconnected (or not necessarily connected) graphs/objects (in this case, bipartite graphs of the nilpotent transformations). The transforms also relate formal cumulants and moments so that Nr(x,t) is then the e.g.f. for the formal moments associated to the formal cumulants whose e.g.f. is T(x,t). (End)
%C A141618 T(n,k) is the number of parking functions of length n containing exactly k+1 distinct values in its image. - _Alan Kappler_, Jun 08 2024
%H A141618 A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1081/AGB-120038637">On the number of nilpotents in the partial symmetric semigroup</a>, Comm. Algebra 32 (2004), 3017-3023.
%H A141618 A. Laradji and A. Umar, <a href="https://mathfiles.kfupm.edu.sa/data/files/mathonly/TechnicalReportsData/305.pdf">On the number of nilpotents in the partial symmetric semigroup</a>, Tech. Report TR305, King Fahd Univ. of Petroleum and Minerals, (2003).
%H A141618 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cumulant#Cumulants_of_some_discrete_probability_distributions">Cumulant</a>
%F A141618 N(J(n,r)) = C(n,r)*S(n,r+1)*r! where S(n, r + 1) is a Stirling number of the second kind (given by A048993 with zeros removed); generating function = (x+1)^(n-1).
%F A141618 From _Peter Bala_, Oct 22 2008: (Start)
%F A141618 Define a functional I on formal power series of the form f(x) = 1 + ax + bx^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim_{n -> infinity} f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).
%F A141618 Let f(x) = 1 + a*x + a*x^2/2! + a*x^3/3! + ... . Then the e.g.f. for this table is I(f(x)) = 1 + a*x +(a + 2*a^2)*x^2/2! + (a + 9*a^2 + 6*a^3)*x^3/3! + (a + 28*a^2 + 72*a^3 + 24*a^4)*x^4/4! + ... . Note, if we take f(x) = 1 + a*x + a*x^2 + a*x^3 + ... then I(f(x)) is the o.g.f. of the Narayana triangle A001263. (End)
%F A141618 A generator for this array is given by the inverse, g(x,t), of f(x,t)= x/(1 + t * (e^x-1)). Then A248927 gives h(x,t)= x / f(x,t) = 1 + t*(e^x-1)= 1 + t * (x + x^2/2! + x^3/3! + ...) and g(x,t)= x * (1 + t * x + (t + 2 t^2) * x^2/2! + (t + 9 t^2 + 6 t^3) * x^3/3! + ...), so by Bala's arguments A248927 is a refinement of A141618 with row sums A000272. The connection to Narayana numbers is reflected in the relation between A248927 and A134264. See A145271 for more relations that g(x,t) and f(x,t) must satisfy. - _Tom Copeland_, Oct 17 2014
%F A141618 T(n,k) = C(n,k-1) * A028246(n,k) = C(n,k-1) * A019538(n,k)/k = A055302(n+1,n+1-k) / (n+1). - _Tom Copeland_, Oct 25 2014
%F A141618 E.g.f. is the series reversion of log(1 + x)/(1 + t*x) with respect to x. Cf. A198204. - _Peter Bala_, Oct 21 2015
%e A141618 N(J(4,2)) = 6*6*2 = 72.
%e A141618 From _Peter Bala_, Oct 22 2008: (Start)
%e A141618 Triangle begins
%e A141618 n\k|..0.....1.....2.....3.....4....5
%e A141618 =====================================
%e A141618 .1.|..1
%e A141618 .2.|..1.....2
%e A141618 .3.|..1.....9.....6
%e A141618 .4.|..1....28....72....24
%e A141618 .5.|..1....75...500...600...120
%e A141618 .6.|..1...186..2700..7800..5400...720
%e A141618 ...
%e A141618 (End)
%p A141618 A048993 := proc(n,k)
%p A141618     combinat[stirling2](n,k) ;
%p A141618 end proc:
%p A141618 A141618 := proc(n,k)
%p A141618     binomial(n,k)*k!*A048993(n,k+1) ;
%p A141618 end proc:
%t A141618 Flatten[CoefficientList[CoefficientList[InverseSeries[Series[Log[1 + x]/(1 + t*x),{x,0,9}]],x]*Table[n!, {n,0,9}],t]] (* _Peter Luschny_, Oct 24 2015, after _Peter Bala_ *)
%o A141618 (PARI)
%o A141618 A055302(n,k)=n!/k!*stirling(n-1, n-k,2);
%o A141618 T(n,k)=A055302(n+1,n+1-k) / (n+1);
%o A141618 for(n=1,10,for(k=1,n,print1(T(n,k),", "));print());
%o A141618 \\ _Joerg Arndt_, Oct 27 2014
%Y A141618 Cf. A000272, A007318, A036040, A048993, A055302, A058877, A127671, A198204.
%K A141618 nonn,tabl
%O A141618 1,3
%A A141618 _Abdullahi Umar_, Aug 23 2008
%E A141618 More terms from _Joerg Arndt_, Oct 27 2014