Please help me with this question. I am stuck. This is a discrete mathematic question. PLEASE PLEASE HELP ME. Recall that \( S(n, k) \) is the Stirling number of the second kind with parameters \( n \) and \( k \).Prove the following:\[\forall n \in \mathbb{Z}^+ \left( S(n, n-1) = C(n, 2) \right).\]

Name: Please help me with this question. I am stuck. This is a discrete math
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please help me with this question. I am stuck. This is a discrete mathematic question. PLEASE PLEASE HELP ME. Recall that \( S(n, k) \) is the Stirling number of the second kind with parameters \( n \) and \( k \).Prove the following:\[\forall n \in

Let ∑n,n-1 denotes the set of all partitions of n into n-1 parts. Now, any element of ∑n,n-1 can be expressed in the form B1⊔B2⊔.....⊔Bn-1 here, Bi is non empty set for every i. Now, take bi=Bi then bi≥1 and ∑i=1n-1bi=n Now, if there are at least two indices say i, j such that bi, bj⩾2 Then n=∑i=1n-1bi ≥2+2+n-1-2 =n+1 which is a contradiction. On the other hand , if bi=1 ∀i Then we get the equation n=n-1 Thus there is one index i such that bi=2 and bj=1 ∀j≠iNow let's assume i=n-1 This shows that an element of ∑n,n-1 is determined by 2 set Bn-i. Conversely, Every 2-subset T of n determines as partitions of n into n-1 blocks n=⊔b∉Tb ⊔T Therefore, we conclude that, Sn,n-1=n2 or Cn, 2 , n∈Z+