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Number of partitions of [n] having exactly one block of maximal size and one block of minimal size (and any number of non-extremal blocks).
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%I #13 May 13 2024 15:13:26

%S 0,1,1,4,5,16,82,169,1381,4162,34346,109099,1114610,5041271,39441963,

%T 269812729,1972727781,14983080612,126099739072,989666749503,

%U 8839669627570,79767000198673,725399587976669,6979798715649335,69812296785011890,703554021895986941

%N Number of partitions of [n] having exactly one block of maximal size and one block of minimal size (and any number of non-extremal blocks).

%C Minimal block and maximal block are identical if there is only one block.

%H Alois P. Heinz, <a href="/A372802/b372802.txt">Table of n, a(n) for n = 0..577</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e a(1) = 1: 1.

%e a(2) = 1: 12.

%e a(3) = 4: 123, 12|3, 13|2, 1|23.

%e a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.

%e a(5) = 16: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.

%e a(8) = 1381: 12345678, 1234567|8, 1234568|7, ..., 1|27|38|456, 18|2|37|456, 1|28|37|456.

%p b:= proc(n, i) option remember; `if`(n=i, 1, 0)+`if`(i<n, add(

%p b(n-i*j, i+1)*combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i), 0)

%p end:

%p a:= n-> signum(n)+add(binomial(n,i)*b(n-i, i+1), i=1..(n-1)/2):

%p seq(a(n), n=0..30);

%Y Cf. A000110, A007837, A224219, A372721.

%K nonn

%O 0,4

%A _Alois P. Heinz_, May 13 2024