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 A319298 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows. 14
 1, 3, 1, 7, 7, 1, 21, 25, 13, 1, 66, 101, 71, 21, 1, 258, 366, 396, 166, 31, 1, 1079, 1555, 1877, 1247, 337, 43, 1, 4987, 7099, 9199, 7855, 3305, 617, 57, 1, 25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1, 136723, 184033, 253108, 284968, 203278, 79756, 16126, 1666, 91, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Alois P. Heinz, Rows n = 1..141, flattened Wikipedia, Partition of a set EXAMPLE The 5 set partitions of {1,2,3} are: 1 |2 |3 1 |23 2 |13 3 |12 123 so there are 7 elements in the first (smallest) blocks, 7 in the second blocks and only 1 in the third blocks. Triangle T(n,k) begins: 1; 3, 1; 7, 7, 1; 21, 25, 13, 1; 66, 101, 71, 21, 1; 258, 366, 396, 166, 31, 1; 1079, 1555, 1877, 1247, 337, 43, 1; 4987, 7099, 9199, 7855, 3305, 617, 57, 1; 25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1; ... MAPLE b:= proc(n, l) option remember; `if`(n=0, add(l[i]* x^i, i=1..nops(l)), add(binomial(n-1, j-1)* b(n-j, sort([l[], j])), j=1..n)) end: T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])): seq(T(n), n=1..12); # second Maple program: b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0, add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial](n, i\$j, n-i*j)), j=0..n/i))) end: T:= (n, k)-> b(n, 1, k)[2]: seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Mar 02 2020 MATHEMATICA b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[i]] x^i, {i, 1, Length[l]}], Sum[ Binomial[n-1, j-1] b[n-j, Sort[Append[l, j]]], {j, 1, n}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *) CROSSREFS Column k=1-10 gives A097147, A332942, A332943, A332944, A332945, A332946, A332947, A332948, A332949, A332950. Row sums give A070071. Cf. A319375, A322383. Sequence in context: A132307 A188463 A359576 * A101748 A058606 A135284 Adjacent sequences: A319295 A319296 A319297 * A319299 A319300 A319301 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 07 2018 STATUS approved

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Last modified January 29 13:51 EST 2023. Contains 359923 sequences. (Running on oeis4.)