• Non ci sono risultati.

# Lethn k i be the unsigned Stirling numbers of the first kind

N/A
N/A
Protected

Academic year: 2021

Condividi "Lethn k i be the unsigned Stirling numbers of the first kind"

Copied!
1
0
0

Testo completo

(1)

Problem 11545

(American Mathematical Monthly, Vol.118, January 2011) Proposed by Manuel Kauers (Austria) and Sheng-Lan Ko(Taiwan).

Find a closed-form expression for

n

X

k=0

(−1)k

 2n n + k



s(n + k, k)

where s refers to the (signed) Stirling numbers of the first kind.

Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.

Lethn

k

i

be the unsigned Stirling numbers of the first kind. We recall that hn

k

i

counts the number of permutations of {1, 2, . . . , n} with k cycles. Since s(n, k) = (−1)n−khn

k

i

, it follows that

an:=

n

X

k=0

(−1)k

 2n n + k



s(n + k, k) =

n

X

k=0

(−1)n−k

 2n n − k

  n + k k



=

n

X

k=0

(−1)k2n k

  2n − k n − k

 .

The last sum can be seen as an application of the Inclusion-Exclusion Principle where the product

2n k

  2n − k n − k



counts the number of permutations of the set {1, 2, . . . , 2n} with n cycles where at least k of them are of length 1 (we choose k numbers in 2nk ways and then we complete the permutation inh2n − k

n − k

i ways). Hence an is the number of permutations of the set {1, 2, . . . , 2n} with n cycles all of length greater than 1, that is with all cycles of length 2. Therefore

an =2n n

 n!

2n =

n

Y

k=1

(2k − 1) = (2n − 1)!!.



Riferimenti

Documenti correlati

determinati campioni permette la misurazione del loro colore percepito in termini assoluti, così come il calcolo della differenza di colore tra gli stessi, il suo uso può

In particular, we considered all the expres- sions in an Italian analysis book (Pandolfi, 2013). By using this corpus of expressions and by assum- ing that all numbers and variables

The method presented in this article is based on three core algorithmic ideas: mean shift clustering to detect soma centers (Section 2.2), supervised semantic deconvolution by means

Theorem 10. Every 2-edge-connected graph has an edge-decomposition into connected, countable, cut-faithful graphs... a vertex in J is a cycle in C); two vertices in J are joined by

Even if the dating rule for classical cycles uses growth rates of economic activity, it has been observed (Stock and Watson, 1999) that in a trending series: (i) classical cycle

In social movement study, research on protest events developed around the concept of cycle of protest, under the assumption that, as political opportunities open up, caus-

[r]

Al ritorno negli Stati Uniti, nel 1874, fu accolta con grande ammirazione sia a Boston che a Filadelfia, ricevendo applausi da molta parte della critica dell’epoca; Henry