Question

6. Let S be a given set. If, for some k > 0, S1, S2, ..., Sk are mutually exclusive nonempty subsets of S such that ?_{i=1}^k Si = S, then we call the set {S1, S2, ..., Sk} a partition of S. Let Tn denote the number of different partitions of {1, 2, ..., n}. Thus, T1 = 1 (the only partition being S1 = {1}) and T2 = 2 (the two partitions being {{1, 2}}, {{1}, {2}}). (a) Show, by computing all partitions, that T3 = 5, T4 = 15. (b) Show that Tn+1 = 1 + ?_{k=1}^n (n choose k) Tk and use this equation to compute T10. Hint: One way of choosing a partition of n + 1 items is to call one of the items special. Then we obtain different partitions by first choosing k, k = 0, 1, ..., n, then a subset of size n - k of the non-special items, and then any of the Tk partitions of the remaining k nonspecial items. By adding the special item to the subset of size n - k, we obtain a partition of all n + 1 items.

          6. Let S be a given set. If, for some k > 0, S1, S2, ..., Sk are mutually exclusive nonempty
subsets of S such that ?_{i=1}^k Si = S, then we call the set {S1, S2, ..., Sk} a partition of S. Let Tn denote the number of different partitions of {1, 2, ..., n}. Thus, T1 = 1 (the only partition being S1 = {1}) and T2 = 2 (the two partitions being {{1, 2}}, {{1}, {2}}).
(a) Show, by computing all partitions, that T3 = 5, T4 = 15.
(b) Show that
Tn+1 = 1 + ?_{k=1}^n (n choose k) Tk
and use this equation to compute T10.
Hint: One way of choosing a partition of n + 1 items is to call one of the items special. Then we obtain different partitions by first choosing k, k = 0, 1, ..., n, then a subset of size n - k of the non-special items, and then any of the Tk partitions of the remaining k nonspecial items. By adding the special item to the subset of size n - k, we obtain a partition of all n + 1 items.

6. Let S be a given set. If, for some k > 0, S1, S2, ..., Sk are mutually exclusive nonempty
subsets of S such that ?i=1^k Si = S, then we call the set S1, S2, ..., Sk a partition of S. Let Tn denote the number of different partitions of 1, 2, ..., n. Thus, T1 = 1 (the only partition being S1 = 1) and T2 = 2 (the two partitions being 1, 2, 1, 2).
(a) Show, by computing all partitions, that T3 = 5, T4 = 15.
(b) Show that
Tn+1 = 1 + ?k=1^n (n choose k) Tk
and use this equation to compute T10.
Hint: One way of choosing a partition of n + 1 items is to call one of the items special. Then we obtain different partitions by first choosing k, k = 0, 1, ..., n, then a subset of size n - k of the non-special items, and then any of the Tk partitions of the remaining k nonspecial items. By adding the special item to the subset of size n - k, we obtain a partition of all n + 1 items.

Added by David C.

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