A157400 -id:A157400 - 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)

A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 9, 8, 6, 1, 25, 40, 30, 24, 1, 75, 200, 180, 144, 120, 1, 231, 980, 1260, 1008, 840, 720, 1, 763, 5152, 8820, 8064, 6720, 5760, 5040, 1, 2619, 28448, 61236, 72576, 60480, 51840, 45360, 40320, 1, 9495, 162080, 461160, 653184, 604800, 518400, 453600, 403200, 362880

Offset: 1

Dan Dima, Mar 01 2007

Sum of the n-th row is the number of all permutations of n elements: Sum_{k=1..n, T(n,k)} = n! = A000142(n) We can extend T(n,k)=0, if k<=0 or k>n.
From Peter Luschny, Mar 07 2009: (Start)
Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A102189.
Same partition product with length statistic is A008275.
Diagonal a(A000217(n)) = rising_factorial(1,n-1), A000142(n-1) (n > 0).
Row sum is A000142. (End)
Let k in {1,2,3,...} index the family of sequences A000012, A000085, A057693, A070945, A070946, A070947, ... respectively. Column k is the k-th sequence minus its immediate predecessor. For example, T(5,3)=A057693(5)-A000085(5). - Geoffrey Critzer, May 23 2009

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   3,    2;
  1,   9,    8,    6;
  1,  25,   40,   30,   24;
  1,  75,  200,  180,  144,  120;
  1, 231,  980, 1260, 1008,  840,  720;
  1, 763, 5152, 8820, 8064, 6720, 5760, 5040;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
S. W. Golomb and P. Gaal, On the number of permutations of n objects with greatest cycle length k, Adv. in Appl. Math., 20(1), 1998, 98-107.
IBM Research, Ponder This, December 2006.
Peter Luschny, Counting with Partitions. [From _Peter Luschny_, Mar 07 2009]
Peter Luschny, Generalized Stirling_1 Triangles. [From _Peter Luschny_, Mar 07 2009]
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Cf. A000142.
Cf. A071007, A080510, A028418.
Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395. - Peter Luschny, Mar 07 2009
T(2n,n) gives A052145 (for n>0). - Alois P. Heinz, Apr 21 2017

A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*A(n-j,k), j=1..k)))
    end:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n,k), k=1..n), n=1..10);  # Alois P. Heinz, Feb 11 2013

Table[CoefficientList[ Series[(Exp[x^m/m] - 1) Exp[Sum[x^k/k, {k, 1, m - 1}]], {x, 0, 8}], x]*Table[n!, {n, 0, 8}], {m, 1, 8}] // Transpose // Grid (* Geoffrey Critzer, May 23 2009 *)

def A126074(n, k):
    f = factorial(n)
    P = Partitions(n, max_part=k, inner=[k])
    return sum(f // p.aut() for p in P)
for n in (1..9): print([A126074(n,k) for k in (1..n)]) # Peter Luschny, Apr 17 2016

T(n,1) = 1.
T(n,2) = n! * Sum_{k=1..[n/2]} 1/(k! * (2!)^k * (n-2*k)!).
T(n,k) = n!/k * (1-1/(n-k)-...-1/(k+1)-1/2k), if n/3 < k <= n/2.
T(n,k) = n!/k, if n/2 < k <= n.
T(n,n) = (n-1)! = A000142(n-1).
E.g.f. for k-th column: exp(-x^k*LerchPhi(x,1,k))*(exp(x^k/k)-1)/(1-x). - Vladeta Jovovic, Mar 03 2007
From Peter Luschny, Mar 07 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-2}(j-n+1). (End)
Sum_{k=1..n} k * T(n,k) = A028418(n). - Alois P. Heinz, May 17 2016

A157394 A partition product of Stirling_1 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 12, 1, 72, 48, 24, 1, 280, 600, 120, 24, 1, 1740, 4560, 1800, 144, 0, 1, 8484, 40740, 21000, 2520, 0, 0, 1, 57232, 390432, 223440, 33600, 0, 0, 0, 1, 328752, 3811248, 2845584, 438480, 0, 0, 0, 0, 1, 2389140

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144878.
Same partition product with length statistic is A049424.
Diagonal a(A000217(n)) = falling_factorial(4,n-1), row in A008279
Row sum is A049427.

			1
1       4
1      12       12
1      72       48       24
1     280      600      120      24
1    1740     4560     1800     144  0
1    8484    40740    21000    2520  0  0
1   57232   390432   223440   33600  0  0  0
1  328752  3811248  2845584  438480  0  0  0  0
1  2389140

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

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395

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-2}(j-n+6).

A157384 A partition product of Stirling_1 type [parameter k = -4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 20, 1, 72, 80, 120, 1, 280, 1000, 600, 840, 1, 1740, 9200, 9000, 5040, 6720, 1, 8484, 79100, 138600, 88200, 47040, 60480, 1, 57232, 874720, 1789200, 1552320, 940800, 483840, 604800, 1, 328752, 9532880

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144354.
Same partition product with length statistic is A049352.
Diagonal a(A000217(n)) = rising_factorial(4,n-1), A001715(n+2).
Row sum is A049377.

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

Cf. A157386, A157384, A157383, A157400, A126074, A157391, A157392, A157393, A157394, A157395

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-2}(j-n-2).

A157391 A partition product of Stirling_1 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 1, 1, 3, 0, 1, 9, 0, 0, 1, 25, 0, 0, 0, 1, 75, 0, 0, 0, 0, 1, 231, 0, 0, 0, 0, 0, 1, 763, 0, 0, 0, 0, 0, 0, 1, 2619, 0, 0, 0, 0, 0, 0, 0, 1, 9495, 0, 0, 0, 0, 0, 0, 0, 0, 1, 35695, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 140151

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 1,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144357.
Same partition product with length statistic is A049403.
Diagonal a(A000217(n)) = falling_factorial(1,n-1), row in A008279.
Row sum is A000085.

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

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395

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-2}(j-n+3).

A157392 A partition product of Stirling_1 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 24, 8, 0, 1, 80, 60, 0, 0, 1, 330, 320, 0, 0, 0, 1, 1302, 2030, 0, 0, 0, 0, 1, 5936, 12432, 0, 0, 0, 0, 0, 1, 26784, 81368, 0, 0, 0, 0, 0, 0, 1, 133650, 545120, 0, 0, 0, 0, 0, 0, 0, 1, 669350, 3825690

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 2,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144358.
Same partition product with length statistic is A049404.
Diagonal a(A000217(n)) = falling_factorial(2,n-1), row in A008279
Row sum is A049425.

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

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395

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-2}(j-n+4).

A157393 A partition product of Stirling_1 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 3, 1, 9, 6, 1, 45, 24, 6, 1, 165, 240, 30, 0, 1, 855, 1560, 360, 0, 0, 1, 3843, 12180, 3360, 0, 0, 0, 1, 21819, 96096, 30660, 0, 0, 0, 0, 1, 114075, 794304, 318276, 0, 0, 0, 0, 0, 1, 703215, 6850080, 3270960, 0, 0, 0, 0, 0, 0, 1, 4125495, 62516520, 35053920, 0, 0

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product prod_{j=0..n-2}(k-n+j+2) and n! at k = 3, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144877.
Same partition product with length statistic is A049410.
Diagonal a(A000217(n)) = falling_factorial(3,n-1), row in A008279.
Row sum is A049426.

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

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395.

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-2}(j-n+5).

A157395 A partition product of Stirling_1 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 20, 1, 105, 80, 60, 1, 425, 1200, 300, 120, 1, 3075, 10400, 5400, 720, 120, 1, 15855, 102200, 75600, 15120, 840, 0, 1, 123515, 1149120, 907200, 241920, 20160, 0, 0, 1, 757755, 12783680, 13426560, 3719520, 362880

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144879.
Same partition product with length statistic is A049411.
Diagonal a(A000217(n)) = falling_factorial(5,n-1), row in A008279
Row sum is A049428.

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

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395

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-2}(j-n+7).

A157383 A partition product of Stirling_1 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 3, 1, 9, 12, 1, 45, 48, 60, 1, 165, 480, 300, 360, 1, 855, 3840, 3600, 2160, 2520, 1, 3843, 29400, 46200, 30240, 17640, 20160, 1, 21819, 272832, 520800, 443520, 282240, 161280, 181440

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -3,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144353.
Same partition product with length statistic is A046089.
Diagonal a(A000217(n)) = rising_factorial(3,n-1), A001710(n+1).
Row sum is A049376.

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

Cf. A157386, A157385, A157384, A157400, A126074, A157391, A157392, A157393, A157394, A157395

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-2}(j-n-1).

A157386 A partition product of Stirling_1 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 6, 1, 18, 42, 1, 144, 168, 336, 1, 600, 2940, 1680, 3024, 1, 4950, 33600, 35280, 18144, 30240, 1, 26586, 336630, 717360, 444528, 211680, 332640, 1, 234528, 4870992, 11313120, 10329984, 5927040, 2661120, 3991680

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144356.
Same partition product with length statistic is A049374.
Diagonal a(A000217(n)) = rising_factorial(6,n-1), A001725(n+4).
Row sum is A049402.

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

Cf. A157385, A157384, A157383, A157400, A126074, A157391, A157392, A157393, A157394, A157395

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-2}(j-n-4).

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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A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

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A157394 A partition product of Stirling_1 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

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A157384 A partition product of Stirling_1 type [parameter k = -4] with biggest-part statistic (triangle read by rows).

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A157391 A partition product of Stirling_1 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

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A157392 A partition product of Stirling_1 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

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A157393 A partition product of Stirling_1 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

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A157395 A partition product of Stirling_1 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

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A157383 A partition product of Stirling_1 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

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