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A271424 Number T(n,k) of set partitions of [n] with minimal block length multiplicity equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A271424 #22 Oct 18 2018 16:20:48
%S A271424 1,0,1,0,1,1,0,4,0,1,0,11,3,0,1,0,51,0,0,0,1,0,132,55,15,0,0,1,0,771,
%T A271424 105,0,0,0,0,1,0,3089,945,0,105,0,0,0,1,0,18388,1218,1540,0,0,0,0,0,1,
%U A271424 0,96423,15456,3150,0,945,0,0,0,0,1,0,627529,26785,24255,0,0,0,0,0,0,0,1
%N A271424 Number T(n,k) of set partitions of [n] with minimal block length multiplicity equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A271424 At least one block length occurs exactly k times, and all block lengths occur at least k times.
%H A271424 Alois P. Heinz, <a href="/A271424/b271424.txt">Rows n = 0..140, flattened</a>
%H A271424 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A271424 T(4,1) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
%e A271424 T(4,2) = 3: 12|34, 13|24, 14|23.
%e A271424 T(4,4) = 1: 1|2|3|4.
%e A271424 T(6,3) = 15: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 13|25|46, 13|26|45, 14|23|56, 15|23|46, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
%e A271424 Triangle T(n,k) begins:
%e A271424   1;
%e A271424   0,     1;
%e A271424   0,     1,     1;
%e A271424   0,     4,     0,    1;
%e A271424   0,    11,     3,    0,   1;
%e A271424   0,    51,     0,    0,   0,   1;
%e A271424   0,   132,    55,   15,   0,   0, 1;
%e A271424   0,   771,   105,    0,   0,   0, 0, 1;
%e A271424   0,  3089,   945,    0, 105,   0, 0, 0, 1;
%e A271424   0, 18388,  1218, 1540,   0,   0, 0, 0, 0, 1;
%e A271424   0, 96423, 15456, 3150,   0, 945, 0, 0, 0, 0, 1;
%p A271424 with(combinat):
%p A271424 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A271424       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
%p A271424         *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
%p A271424     end:
%p A271424 T:= (n, k)-> b(n$2, k)-`if`(n=k,0,b(n$2, k+1)):
%p A271424 seq(seq(T(n, k), k=0..n), n=0..12);
%t A271424 multinomial[n_, k_List] := 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, Join[{0}, Range[k, n/i]] // Union}]]]; T[n_, k_] := b[n, n, k] - If[n == k, 0, b[n, n, k + 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 16 2017, adapted from Maple *)
%Y A271424 Columns k=0-10 give A000007, A271426, A271762, A271763, A271764, A271765, A271766, A271767, A271768, A271769, A271770.
%Y A271424 Row sums give A000110.
%Y A271424 Main diagonal gives A000012.
%Y A271424 T(2n,n) gives A001147.
%Y A271424 T(3n,n) gives A271715.
%Y A271424 Cf. A271423.
%K A271424 nonn,tabl
%O A271424 0,8
%A A271424 _Alois P. Heinz_, Apr 07 2016