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 A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k. 33
 1, 1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 25, 20, 5, 1, 1, 75, 90, 30, 6, 1, 1, 231, 420, 175, 42, 7, 1, 1, 763, 2016, 1015, 280, 56, 8, 1, 1, 2619, 10024, 6111, 1890, 420, 72, 9, 1, 1, 9495, 51640, 38010, 12978, 3150, 600, 90, 10, 1, 1, 35695, 276980, 244035, 91938, 24024, 4950, 825, 110, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are A000110 (Bell numbers). Second column is A001189 (Degree n permutations of order exactly 2). From Peter Luschny, Mar 09 2009: (Start) Partition product of Product_{j=0..n-1} ((k + 1)*j - 1) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link). Underlying partition triangle is A036040. Same partition product with length statistic is A008277. Diagonal a(A000217) = A000012. Row sum is A000110. (End) From Gary W. Adamson, Feb 24 2011: (Start) Construct an array in which the n-th row is the partition function G(n,k), where G(n,1),...,G(n,6) = A000012, A000085, A001680, A001681, A110038, A148092, with the first few rows 1, 1, 1, 1, 1, 1, 1, ... = A000012 1, 2, 4, 10, 26, 76, 232, ... = A000085 1, 2, 5, 14, 46, 166, 652, ... = A001680 1, 2, 5, 15, 51, 196, 827, ... = A001681 1, 2 5 15 52 202 869, ... = A110038 1, 2, 5 15 52 203 876, ... = A148092 ... Rows tend to A000110, the Bell numbers. Taking finite differences from the top, then reorienting, we obtain triangle A080510. The n-th row of the array is the eigensequence of an infinite lower triangular matrix with n diagonals of Pascal's triangle starting from the right and the rest zeros. (End) LINKS Alois P. Heinz, Rows n = 1..141, flattened Peter Luschny, Counting with Partitions. Peter Luschny, Generalized Stirling_2 Triangles. J. Riordan, Letter, 11/23/1970. See second page of letter. FORMULA E.g.f. for k-th column: exp(exp(x)*GAMMA(k, x)/(k-1)!-1)*(exp(x^k/k!)-1). - Vladeta Jovovic, Feb 04 2005 From Peter Luschny, Mar 09 2009: (Start) T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n. T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,...,a_n such that 1*a_1 + 2*a_2 + ... + n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*...*a_n!), f^a = (f_1/1!)^a_1*...*(f_n/n!)^a_n and f_n = Product_{j=0..n-1} (-1) = (-1)^n. (End) From Ludovic Schwob, Jan 15 2022: (Start) T(2n,n) = C(2n,n)*(A000110(n)-1/2) for n>0. T(n,m) = C(n,m)*A000110(n-m) for 2m > n > 0. (End) EXAMPLE T(4,3) = 4 since there are 4 set partitions with longest block of length 3: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}} and {{1,2,4},{3}}. Triangle begins: 1; 1, 1; 1, 3, 1; 1, 9, 4, 1; 1, 25, 20, 5, 1; 1, 75, 90, 30, 6, 1; 1, 231, 420, 175, 42, 7, 1; 1, 763, 2016, 1015, 280, 56, 8, 1; 1, 2619, 10024, 6111, 1890, 420, 72, 9, 1; ... MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i))) end: T:= (n, k)-> b(n, k) -b(n, k-1): seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Apr 20 2012 MATHEMATICA << DiscreteMath`NewCombinatorica`; Table[Length/@Split[Sort[Max[Length/@# ]&/@SetPartitions[n]]], {n, 12}] (* Second program: *) b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; T[n_, k_] := b[n, k]-b[n, k-1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *) CROSSREFS Cf. A080107, A080337, A008277, A178979, A276922, A327884. Cf. A157396, A157397, A157398, A157399, A157400, A157401, A157402, A157403, A157404, A157405. - Peter Luschny, Mar 09 2009 Cf. A000012, A000085, A001680, A001681, A110038, A148092. - Gary W. Adamson, Feb 24 2011 Columns k=1..10 give: A000012 (for n>0), A001189, A229245, A229246, A229247, A229248, A229249, A229250, A229251, A229252. - Alois P. Heinz, Sep 17 2013 T(2n,n) gives A276961. Take differences along rows of A229223. - N. J. A. Sloane, Jan 10 2018 Sequence in context: A152570 A100537 A069605 * A350772 A350783 A124496 Adjacent sequences: A080507 A080508 A080509 * A080511 A080512 A080513 KEYWORD nonn,tabl AUTHOR Wouter Meeussen, Mar 22 2003 STATUS approved

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Last modified February 29 00:33 EST 2024. Contains 370400 sequences. (Running on oeis4.)