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.

1, 1, 2, 1, 9, 6, 1, 28, 72, 24, 1, 75, 500, 600, 120, 1, 186, 2700, 7800, 5400, 720, 1, 441, 12642, 73500, 117600, 52920, 5040, 1, 1016, 54096, 571536, 1764000, 1787520, 564480, 40320, 1, 2295, 217800, 3916080, 21019824, 40007520, 27941760, 6531840, 362880, 1, 5110, 839700, 24555600, 214326000

Offset

1,3

Comments

The sum of each row of the sequence (as a triangular array) is A000272. Second left-downward diagonal is A058877.

From Tom Copeland, Oct 26 2014: (Start)

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).

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).

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)].

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)

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

Links

Table of n, a(n) for n=1..50.

A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.

A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Tech. Report TR305, King Fahd Univ. of Petroleum and Minerals, (2003).

Wikipedia, Cumulant

Formula

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).

From Peter Bala, Oct 22 2008: (Start)

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).

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)

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

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

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

Example

N(J(4,2)) = 6*6*2 = 72.
From Peter Bala, Oct 22 2008: (Start)
Triangle begins
n\k|..0.....1.....2.....3.....4....5
=====================================
.1.|..1
.2.|..1.....2
.3.|..1.....9.....6
.4.|..1....28....72....24
.5.|..1....75...500...600...120
.6.|..1...186..2700..7800..5400...720
...
(End)

Maple

A048993 := proc(n, k)
    combinat[stirling2](n, k) ;
end proc:
A141618 := proc(n, k)
    binomial(n, k)*k!*A048993(n, k+1) ;
end proc:

Mathematica

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 *)

Prog

(PARI)
A055302(n, k)=n!/k!*stirling(n-1, n-k, 2);
T(n, k)=A055302(n+1, n+1-k) / (n+1);
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print());
\\ Joerg Arndt, Oct 27 2014

Crossrefs

Cf. A000272, A007318, A036040, A048993, A055302, A058877, A127671, A198204.

Sequence in context: A103876 A133174 A155545 * A061691 A235595 A061356

Adjacent sequences: A141615 A141616 A141617 * A141619 A141620 A141621

Keyword

nonn,tabl

Author

Abdullahi Umar, Aug 23 2008

Extensions

More terms from Joerg Arndt, Oct 27 2014

Status

approved