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%I #22 May 14 2023 13:08:25
%S 1,0,1,0,1,1,0,4,0,1,0,5,9,0,1,0,16,25,10,0,1,0,82,70,35,15,0,1,0,169,
%T 406,245,35,21,0,1,0,541,2093,1036,385,56,28,0,1,0,2272,10935,4984,
%U 2331,504,84,36,0,1,0,17966,41961,37990,13335,3717,840,120,45,0,1
%N Number T(n,k) of set partitions of [n] with maximal block length multiplicity equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C At least one block length occurs exactly k times, and all block lengths occur at most k times.
%H Alois P. Heinz, <a href="/A271423/b271423.txt">Rows n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e T(4,1) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
%e T(4,2) = 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 T(4,4) = 1: 1|2|3|4.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 4, 0, 1;
%e 0, 5, 9, 0, 1;
%e 0, 16, 25, 10, 0, 1;
%e 0, 82, 70, 35, 15, 0, 1;
%e 0, 169, 406, 245, 35, 21, 0, 1;
%e 0, 541, 2093, 1036, 385, 56, 28, 0, 1;
%e 0, 2272, 10935, 4984, 2331, 504, 84, 36, 0, 1;
%e 0, 17966, 41961, 37990, 13335, 3717, 840, 120, 45, 0, 1;
%e ...
%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 T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t 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, 0, Min[k, n/i]}]]]; T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 06 2017, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A007837 (for n>0), A271731, A271732, A271733, A271734, A271735, A271736, A271737, A271738, A271739.
%Y Row sums give A000110.
%Y Main diagonal gives A000012.
%Y T(2n,n) gives A271425.
%Y Cf. A271424.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Apr 07 2016
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