A157403 -id:A157403 - cp's OEIS frontend

A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k.

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

Offset: 1

Wouter Meeussen, Mar 22 2003

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)

			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;
  ...

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.

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

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

<< 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 *)

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)

A157400 A partition product with biggest-part statistic of Stirling_1 type (with parameter k = -2) as well as of Stirling_2 type (with parameter k = -2), (triangle read by rows).

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 24, 24, 24, 1, 80, 180, 120, 120, 1, 330, 1200, 1080, 720, 720, 1, 1302, 7770, 10920, 7560, 5040, 5040, 1, 5936, 57456, 102480, 87360, 60480, 40320, 40320, 1, 26784, 438984, 970704, 1103760, 786240, 544320, 362880, 362880

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of Product_{j=0..n-1} ((k+1)*j - 1) and n! at k = -2, summed over parts with equal biggest part (Stirling_2 type) as well as partition product of Product_{j=0..n-2} (k-n+j+2) and n! at k = -2 (Stirling_1 type).
It shares this property with the signless Lah numbers.
Underlying partition triangle is A130561.
Same partition product with length statistic is A105278.
Diagonal a(A000217) = A000142.
Row sum is A000262.
T(n,k) is the number of nilpotent elements in the symmetric inverse semigroup (partial bijections) on [n] having index k. Equivalently, T(n,k) is the number of directed acyclic graphs on n labeled nodes with every node having indegree and outdegree at most one and the longest path containing exactly k nodes. - Geoffrey Critzer, Nov 21 2021

			Triangle starts:
  1;
  1,   2;
  1,   6,    6;
  1,  24,   24,   24;
  1,  80,  180,  120, 120;
  1, 330, 1200, 1080, 720, 720;
  ...

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_1 Triangles.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A080510, A157401, A157402, A157403, A157404, A157405, A157386, A157385, A157384, A157383, A126074, A157391, A157392, A157393, A157394, A157395.

egf:= k-> exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)):
T:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Oct 10 2015

egf[k_] := Exp[(x^(k+1)-x)/(x-1)] - Exp[(x^k-x)/(x-1)]; T[n_, k_] := n! * SeriesCoefficient[egf[k], {x, 0, n}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Oct 11 2015, after Alois P. Heinz *)

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} (-j-1)
OR f_n = Product_{j=0..n-2} (j-n) since both have the same absolute value n!.
E.g.f. of column k: exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)). - Alois P. Heinz, Oct 10 2015

A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 6, 1, 18, 66, 1, 144, 264, 1056, 1, 600, 4620, 5280, 22176, 1, 4950, 68640, 110880, 133056, 576576, 1, 26586, 639870, 3141600, 3259872, 4036032, 17873856, 1, 234528, 10759056, 69263040, 105557760, 113008896, 142990848

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134278.
Same partition product with length statistic is A049385.
Diagonal a(A000217) = A008548.
Row sum is A049412.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404, A157405

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}(-5*j - 1).

Offset corrected by Peter Luschny, Mar 14 2009

A157397 A partition product of Stirling_2 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 45, 1, 105, 180, 585, 1, 425, 2700, 2925, 9945, 1, 3075, 34650, 52650, 59670, 208845, 1, 15855, 308700, 1248975, 1253070, 1461915, 5221125, 1, 123515, 4475520, 23689575, 33972120, 35085960, 41769000

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134273.
Same partition product with length statistic is A049029.
Diagonal a(A000217) = A007696.
Row sum is A049120.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404, A157405

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}(-4*j - 1).

Offset corrected by Peter Luschny, Mar 14 2009

A157398 A partition product of Stirling_2 type [parameter k = -4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 28, 1, 72, 112, 280, 1, 280, 1400, 1400, 3640, 1, 1740, 15120, 21000, 21840, 58240, 1, 8484, 126420, 401800, 382200, 407680, 1106560, 1, 57232, 1538208, 6370000, 8357440, 8153600, 8852480, 24344320, 1

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134149.
Same partition product with length statistic is A035469.
Diagonal a(A000217) = A007559.
Row sum is A049119.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157399, A157400, A080510, A157401, A157402, A157403, A157404, A157405

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}(-3*j - 1).

A157399 A partition product of Stirling_2 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 3, 1, 9, 15, 1, 45, 60, 105, 1, 165, 600, 525, 945, 1, 855, 5250, 6300, 5670, 10395, 1, 3843, 39900, 91875, 79380, 72765, 135135, 1, 21819, 391440, 1164975, 1323000, 1164240, 1081080, 2027025, 1

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -3,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134144.
Same partition product with length statistic is A035342.
Diagonal a(A000217) = A001147.
Row sum is A049118.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157400, A080510, A157401, A157402, A157403, A157404, A157405

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}(-2*j - 1).

A157401 A partition product of Stirling_2 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 9, 12, 15, 1, 25, 60, 75, 105, 1, 75, 330, 450, 630, 945, 1, 231, 1680, 3675, 4410, 6615, 10395, 1, 763, 9408, 30975, 41160, 52920, 83160, 135135, 1, 2619, 56952, 233415, 489510, 555660, 748440, 1216215

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{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 A143171.
Same partition product with length statistic is A001497.
Diagonal a(A000217) = A001147.
Row sum is A001515.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157402, A157403, A157404, A157405

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}(2*j - 1).

A157402 A partition product of Stirling_2 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 2, 1, 6, 10, 1, 24, 40, 80, 1, 80, 300, 400, 880, 1, 330, 2400, 3600, 5280, 12320, 1, 1302, 15750, 47600, 55440, 86240, 209440, 1, 5936, 129360, 588000, 837760, 1034880, 1675520, 4188800, 1, 26784, 1146040, 5856480

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 2,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143172.
Same partition product with length statistic is A004747.
Diagonal a(A000217) = A008544.
Row sum is A015735.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157403, A157404, A157405

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}(3*j - 1).

A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 36, 1, 72, 144, 504, 1, 280, 1800, 2520, 9576, 1, 1740, 22320, 37800, 57456, 229824, 1, 8484, 182700, 864360, 1005480, 1608768, 6664896, 1, 57232, 2380896, 16546320, 26276544, 32175360, 53319168, 226606464

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144267.
Same partition product with length statistic is A011801.
Diagonal a(A000217) = A008546.
Row sum is A028575.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157405

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}(5*j - 1).

A157405 A partition product of Stirling_2 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 55, 1, 105, 220, 935, 1, 425, 3300, 4675, 21505, 1, 3075, 47850, 84150, 129030, 623645, 1, 15855, 415800, 2323475, 2709630, 4365515, 415800, 2323475, 2709630, 4365515, 21827575, 1, 123515, 6394080, 51934575

Offset: 0

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144268.
Same partition product with length statistic is A013988.
Diagonal a(A000217) = A008543.
Row sum is A028844.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404

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}(6*j - 1).

cp's OEIS Frontend

A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k.

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A157400 A partition product with biggest-part statistic of Stirling_1 type (with parameter k = -2) as well as of Stirling_2 type (with parameter k = -2), (triangle read by rows).

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A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

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A157397 A partition product of Stirling_2 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

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A157398 A partition product of Stirling_2 type [parameter k = -4] with biggest-part statistic (triangle read by rows).

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A157399 A partition product of Stirling_2 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

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A157401 A partition product of Stirling_2 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

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A157402 A partition product of Stirling_2 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

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A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

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