A271425 - OEIS

%I #13 May 12 2024 14:20:04

%S 1,1,9,35,385,3717,48279,691119,11229075,200982925,3928974907,

%T 83060120871,1885501840677,45694145548625,1176704027583075,

%U 32077561625780175,922854842240358825,27951355368760441365,889580295850449177975,29707539555680924142975

%N Number of set partitions of [2n] with maximal block length multiplicity equal to n.

%C In each set partition of [2n] counted by a(n) at least one block length occurs exactly n times, and all block lengths occur at most n times.

%H Alois P. Heinz, <a href="/A271425/b271425.txt">Table of n, a(n) for n = 0..200</a>

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

%F a(n) = A271423(2n,n).

%F a(n) = A372762(2n,n). - _Alois P. Heinz_, May 12 2024

%e a(1) = 1: 12.

%e a(2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.

%e a(3) = 35: 123|4|5|6, 124|3|5|6, 12|34|56, 125|3|4|6, 12|35|46, 12|36|45, 126|3|4|5, 134|2|5|6, 13|24|56, 135|2|4|6, 13|25|46, 13|26|45, 136|2|4|5, 14|23|56, 1|234|5|6, 15|23|46, 1|235|4|6, 16|23|45, 1|236|4|5, 145|2|3|6, 14|25|36, 14|26|35, 146|2|3|5, 15|24|36, 1|245|3|6, 16|24|35, 1|246|3|5, 15|26|34, 16|25|34, 1|2|345|6, 1|2|346|5, 156|2|3|4, 1|256|3|4, 1|2|356|4, 1|2|3|456.

%p with(combinat):

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

%p       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)

%p         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))

%p     end:

%p a:= n-> `if`(n=0, 1, b(2*n$2, n)-b(2*n$2, n-1)):

%p seq(a(n), n=0..20);

%t multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]*b[n - i*j, i-1, k]/j!, {j, 0, Min[k, n/i]}]]]; a[n_] := If[n==0, 1, b[2n, 2n, n] - b[2n, 2n, n-1]]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 17 2017, translated from Maple *)

%Y Cf. A271423, A372762.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Apr 07 2016