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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. 13
 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, 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, 0, 96423, 15456, 3150, 0, 945, 0, 0, 0, 0, 1, 0, 627529, 26785, 24255, 0, 0, 0, 0, 0, 0, 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 least k times. LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Partition of a set EXAMPLE 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. T(4,2) = 3: 12|34, 13|24, 14|23. T(4,4) = 1: 1|2|3|4. 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. Triangle T(n,k) begins: 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, 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; 0, 96423, 15456, 3150, 0, 945, 0, 0, 0, 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, \$k..n/i}))) end: T:= (n, k)-> b(n\$2, k)-`if`(n=k, 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, 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 *) CROSSREFS Columns k=0-10 give A000007, A271426, A271762, A271763, A271764, A271765, A271766, A271767, A271768, A271769, A271770. Row sums give A000110. Main diagonal gives A000012. T(2n,n) gives A001147. T(3n,n) gives A271715. Cf. A271423. Sequence in context: A363973 A046781 A244530 * A117435 A282252 A268367 Adjacent sequences: A271421 A271422 A271423 * A271425 A271426 A271427 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Apr 07 2016 STATUS approved

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Last modified March 5 07:51 EST 2024. Contains 370538 sequences. (Running on oeis4.)