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 A271423 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. 13
 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, 406, 245, 35, 21, 0, 1, 0, 541, 2093, 1036, 385, 56, 28, 0, 1, 0, 2272, 10935, 4984, 2331, 504, 84, 36, 0, 1, 0, 17966, 41961, 37990, 13335, 3717, 840, 120, 45, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS At least one block length occurs exactly k times, and all block lengths occur at most k times. LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Partition of a set EXAMPLE T(4,1) = 5: 1234, 123|4, 124|3, 134|2, 1|234. 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. T(4,4) = 1: 1|2|3|4. Triangle T(n,k) begins:   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,   406,   245,    35,   21,   0,   1;   0,   541,  2093,  1036,   385,   56,  28,   0,  1;   0,  2272, 10935,  4984,  2331,  504,  84,  36,  0, 1;   0, 17966, 41961, 37990, 13335, 3717, 840, 120, 45, 0, 1; MAPLE with(combinat): b:= proc(n, i, k) option remember; `if`(n=0, 1,       `if`(i<1, 0, add(multinomial(n, n-i*j, i\$j)         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))     end: T:= (n, k)-> b(n\$2, k)-`if`(k=0, 0, b(n\$2, k-1)): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA 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 *) CROSSREFS Columns k=0-10 give: A000007, A007837 (for n>0), A271731, A271732, A271733, A271734, A271735, A271736, A271737, A271738, A271739. Row sums give A000110. Main diagonal gives A000012. T(2n,n) gives A271425. Cf. A271424. Sequence in context: A272774 A147311 A147312 * A019974 A046781 A244530 Adjacent sequences:  A271420 A271421 A271422 * A271424 A271425 A271426 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Apr 07 2016 STATUS approved

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Last modified November 16 12:40 EST 2018. Contains 317272 sequences. (Running on oeis4.)