A056857 -id:A056857 - cp's OEIS frontend

A056860 Triangle T(n,k) = number of element-subset partitions of {1..n} with n-k+1 equalities (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 20, 15, 1, 5, 20, 50, 75, 52, 1, 6, 30, 100, 225, 312, 203, 1, 7, 42, 175, 525, 1092, 1421, 877, 1, 8, 56, 280, 1050, 2912, 5684, 7016, 4140, 1, 9, 72, 420, 1890, 6552, 17052, 31572, 37260, 21147

Offset: 1

N. J. A. Sloane, Oct 13 2000

T(n,k) = number of permutations on [n] with n in position k in which 321 patterns only occur as part of 3241 patterns. Example: T(4,2)=3 counts 1423, 2413, 3412. - David Callan, Jul 20 2005
From Gary W. Adamson, Feb 24 2011: (Start)
Given rows of an array such that n-th row is the eigensequence of an infinite lower triangular matrix with first n columns of Pascal's triangle and the rest zeros. The reoriented finite differences of the array starting from the top are the rows of A056860.
The first few rows of the array are
  1,   1,   1,   1,   1,   1, ...
  1,   2,   3,   4,   5,   6, ...
  1,   2,   5,  10,  17,  26, ...
  1,   2,   5,  15,  37,  76, ...
  1,   2,   5,  15,  52, 151, ...
  ...
(End)

			T(n,k) starts:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  6,   5;
  1, 4, 12,  20,   15;
  1, 5, 20,  50,   75,   52;
  1, 6, 30, 100,  225,  312,   203;
  1, 7, 42, 175,  525, 1092,  1421,   877;
  1, 8, 56, 280, 1050, 2912,  5684,  7016,  4140;
  1, 9, 72, 420, 1890, 6552, 17052, 31572, 37260, 21147;
Building row sums Sum_{c=1..k} T(n,c), the following array results:
  1, 1,  1,   1,    1,    1,    1,     1,     1,     1, ...
  1, 2,  2,   2,    2,    2,    2,     2,     2,     2, ...
  1, 3,  5,   5,    5,    5,    5,     5,     5,     5, ...
  1, 4, 10,  15,   15,   15,   15,    15,    15,    15, ...
  1, 5, 17,  37,   52,   52,   52,    52,    52,    52, ...
  1, 6, 26,  76,  151,  203,  203,   203,   203,   203, ...
  1, 7, 37, 137,  362,  674,  877,   877,   877,   877, ...
  1, 8, 50, 225,  750, 1842, 3263,  4140,  4140,  4140, ...
  1, 9, 65, 345, 1395, 4307, 9991, 17007, 21147, 21147, ...

W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.

David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

Essentially same as A056857, where rows are read from left to right.
T(2n+1,n+1) gives A124102.
T(2n,n) gives A297926.

T(n, k) = binomial(n-1, k-1)*B(k-1) where B denotes the Bell numbers A000110. - David Callan, Jul 20 2005

More terms from David Callan, Jul 20 2005

A259691 Triangle read by rows: T(n,k) number of arrangements of non-attacking rooks on an n X n right triangular board where the top rook is in row k (n >= 0, 1 <= k <= n+1).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 6, 3, 1, 15, 20, 12, 4, 1, 52, 74, 51, 20, 5, 1, 203, 302, 231, 104, 30, 6, 1, 877, 1348, 1116, 564, 185, 42, 7, 1, 4140, 6526, 5745, 3196, 1175, 300, 56, 8, 1, 21147, 34014, 31443, 18944, 7700, 2190, 455, 72, 9, 1

Offset: 0

N. J. A. Sloane, Jul 05 2015

Another version of A056857.
See Becker (1948/49) for precise definition.
The case of n=k+1 corresponds to the empty board where there is no top rook. - Andrew Howroyd, Jun 13 2017
T(n-1,k) is the number of partitions of [n] where exactly k blocks contain their own index element.  T(3,2) = 6: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4. - Alois P. Heinz, Jan 07 2022

			Triangle begins:
    1;
    1,   1;
    2,   2,   1;
    5,   6,   3,   1;
   15,  20,  12,   4,  1;
   52,  74,  51,  20,  5, 1;
  203, 302, 231, 104, 30, 6, 1;
  ...
From _Andrew Howroyd_, Jun 13 2017: (Start)
For n=3 the 5 solutions with the top rook in row 1 are:
  x      x      x      x      x
  . .    . .    . .    . x    . x
  . . .  . . x  . x .  . . .  . . x
For n=3 the 6 solutions with the top rook in row 2 are:
  .      .      .      .      .      .
  x .    x .    x .    . x    . x    . x
  . . .  . x .  . . x  . . .  x . .  . . x
(End)

Alois P. Heinz, Rows n = 0..140, flattened (first 49 rows from Andrew Howroyd)
H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. See Table II.
H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. [Annotated scanned copy]
Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, Set Partitions and Other Bell Number Enumerated Objects, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.
Giulio Cerbai, Modified ascent sequences and Bell numbers, arXiv:2305.10820 [math.CO], 2023. See p. 17.

First column is A000110.
Row sums are A000110(n+1).
Cf. A056857, A259697, A108087, A347420, A350589.

b:= proc(n, m) option remember; `if`(n=0, 1,
     `if`(n<0, 1/m, m*b(n-1, m)+b(n-1, m+1)))
    end:
T:= (n, k)-> k*b(n-k, k):
seq(seq(T(n, k), k=1..n+1), n=0..10);  # Alois P. Heinz, Jan 07 2022

T[n_, k_] := If[k>n, 1, k*Sum[Binomial[n-k, i]*k^i*BellB[n-k-i], {i, 0, n - k}]];
Table[T[n, k], {n, 0, 10}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

bell(n) = sum(k=0, n, stirling(n, k, 2));
T(n,k) = if(k>n, 1, k*sum(i=0,n-k, binomial(n-k,i) * k^i * bell(n-k-i)));
for(n=0,6, for(k=1,n+1, print1(T(n,k),", ")); print) \\ Andrew Howroyd, Jun 13 2017

T(n,n+1) = 1, T(n,k) = k*Sum_{i=0..n-k} binomial(n-k,i) * k^i * Bell(n-k-i) for k<=n. - Andrew Howroyd, Jun 13 2017
From Alois P. Heinz, Jan 07 2022: (Start)
T(n,k) = k * A108087(n-k,k) for 1 <= k <= n.
Sum_{k=1..n+1} k * T(n,k) = A350589(n+1).
Sum_{k=1..n+1} (k+1) * T(n,k) = A347420(n+1). (End)

Name edited and terms a(28) and beyond from Andrew Howroyd, Jun 13 2017

A056863 Related to triangle A056858 of number of rises in set partitions of n at a given index i.

Original entry on oeis.org

1, -1, 1, -2, 1, 1, -3, 4, 2, 1, -4, 9, 10, 4, 1, -5, 16, 28, 24, 8, 1, -6, 25, 60, 80, 56, 16

Offset: 1

Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

			1,-1;
1,-2,1;
1,-3,4,2;
1,-4,9,10,4;
...

W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [Apparently unpublished. - R. J. Mathar, Oct 06 2011]

Cf. Bell numbers A000110.

Cf. A056857-A056862.

A124427 Sum of the sizes of the blocks containing the element 1 in all set partitions of {1,2,...,n}.

Original entry on oeis.org

0, 1, 3, 9, 30, 112, 463, 2095, 10279, 54267, 306298, 1838320, 11677867, 78207601, 550277003, 4055549053, 31224520322, 250547144156, 2090779592827, 18110124715919, 162546260131455, 1509352980864191, 14478981877739094, 143299752100925452, 1461455003961745247

Offset: 0

Emeric Deutsch, Nov 10 2006

nonn

			a(3)=9 because the 5 (=A000110(3)) set partitions of {1,2,3} are 123, 12|3, 13|2, 1|23 and 1|2|3 and 3+2+2+1+1=9.

Alois P. Heinz, Table of n, a(n) for n = 0..575

Cf. A000110, A056857.
Column p=1 of A270236 or of A270702.
Main diagonal of A270701.

with(combinat): seq(add(k*binomial(n-1,k-1)*bell(n-k),k=1..n),n=0..30);

Table[Sum[Binomial[n-1,k-1] * BellB[n-k] * k, {k,1,n}], {n,0,22}] (* Geoffrey Critzer, Jun 14 2013 *)
Flatten[{0, Table[(n-1)*BellB[n-1] + BellB[n], {n, 1, 20}]}] (* Vaclav Kotesovec, Mar 19 2016, after Vladeta Jovovic *)

a(n) = Sum(k*binomial(n-1,k-1)*B(n-k), k=1..n) = Sum(k*A056857(n,k), k=1..n), where B(q) are the Bell numbers (A000110).
a(n) = (n-1)*B(n-1)+B(n). - Vladeta Jovovic, Nov 10 2006
a(n) ~ Bell(n) * (LambertW(n) + 1). - Vaclav Kotesovec, Jul 28 2021

a(0)=0 prepended by Alois P. Heinz, Mar 17 2016

A297926 Number of set partitions of [2n] in which the size of the first block is n.

Original entry on oeis.org

1, 1, 6, 50, 525, 6552, 93786, 1504932, 26640900, 514083570, 10713538550, 239342496120, 5697111804566, 143759365731100, 3829115870472600, 107260549881604200, 3149703964487098665, 96686987797052290440, 3094969650442399156350, 103079905957566679518300

Offset: 0

Alois P. Heinz, Jan 08 2018

nonn

The blocks are ordered with increasing least elements.

a(0) = 1 by convention.

			a(1) = 1: 1|2.
a(2) = 6: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 14|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..445
Wikipedia, Partition of a set

Cf. A000110, A056857, A056860, A276961, A297924.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> binomial(2*n-1, n-1)*b(n):
seq(a(n), n=0..25);

b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := Binomial[2*n-1, n-1] * b[n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

a(n) = binomial(2*n-1,n-1) * Bell(n).

a(n) = A056857(2n,n) = A056860(2n,n).

A175757 Triangular array read by rows: T(n,k) is the number of blocks of size k in all set partitions of {1,2,...,n}.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 20, 12, 4, 1, 75, 50, 20, 5, 1, 312, 225, 100, 30, 6, 1, 1421, 1092, 525, 175, 42, 7, 1, 7016, 5684, 2912, 1050, 280, 56, 8, 1, 37260, 31572, 17052, 6552, 1890, 420, 72, 9, 1, 211470, 186300, 105240, 42630, 13104, 3150, 600, 90, 10, 1

Offset: 1

Geoffrey Critzer, Dec 04 2010

The row sums of this triangle equal A005493. Equals A056857 without its leftmost column.

T(n,k) = binomial(n,k)*B(n-k) where B is the Bell number.

			The set {1,2,3} has 5 partitions, {{1, 2, 3}}, {{2, 3}, {1}}, {{1, 3}, {2}}, {{1, 2}, {3}}, and {{2}, {3}, {1}}, and there are a total of 3 blocks of size 2, so T(3,2)=3.
Triangle begins:
    1;
    2,   1;
    6,   3,   1;
   20,  12,   4,  1;
   75,  50,  20,  5, 1;
  312, 225, 100, 30, 6, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000110, A056860, A052889, A070071, A105479, A105480, A105481, A105482, A285595.

b:= proc(n) option remember; `if`(n=0, [1, 0],
      add((p-> p+[0, p[1]*x^j])(b(n-j)*
      binomial(n-1, j-1)), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)[2]):
seq(T(n), n=1..12);  # Alois P. Heinz, Apr 24 2017

Table[Table[Length[Select[Level[SetPartitions[m],{2}],Length[#]==n&]],{n,1,m}],{m,1,10}]//Grid

E.g.f. for column k is x^k/k!*exp(exp(x)-1).

Sum_{k=1..n} k * T(n,k) = A070071(n). - Alois P. Heinz, Mar 03 2020

A056858 Triangle of number of rises in restricted growth strings (RGS) for the set partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 26, 14, 1, 1, 15, 71, 89, 26, 1, 1, 21, 161, 380, 267, 46, 1, 1, 28, 322, 1268, 1709, 732, 79, 1, 1, 36, 588, 3571, 8136, 6794, 1887, 133, 1, 1, 45, 1002, 8878, 31532, 44924, 24717, 4654, 221, 1, 1, 55, 1617, 20053, 104927, 234412, 221857, 84170, 11113, 364, 1

Offset: 1

Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

Number of rises s_{i+1} > s_i in the RGS [s_1, ..., s_n] for a set partition of {1, ..., n}, where s_i is the index of the subset containing i, s_1 = 1 and s_i <= 1 + max_{j

			For example [1, 2, 1, 2, 2, 3] is the RGS for a set partition of {1, 2, 3, 4, 5, 6} and has 3 rises, at i = 1, i = 3 and i = 5.
1;
1,1;
1,3,1;
1,6,7,1;
1,10,26,14,1;
1,15,71,89,26,1;
1,21,161,380,267,46,1;
1,28,322,1268,1709,732,79,1;
1,36,588,3571,8136,6794,1887,133,1;
1,45,1002,8878,31532,44924,24717,4654,221,1;
1,55,1617,20053,104927,234412,221857,84170,11113,364,1;
1,66,2497,41965,310255,1025377,1528351,1006028,272557,25903,596,1;

W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [apparently unpublished, Joerg Arndt, Mar 05 2016]

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A000110 (row sums).
Cf. A056857-A056863.
Column 1 is triangular numbers (A000217); diagonal T(n, n-1) appears to be A001924.

b:= proc(n, i, m) option remember; expand(
      `if`(n=0, x, add(b(n-1, j, max(m, j))*
      `if`(j>i, x, 1), j=1..m+1)))
    end:
T:= n->(p-> seq(coeff(p, x, i), i=1..n))(b(n, 1, 0)):
seq(T(n), n=1..12);  # Alois P. Heinz, Mar 24 2016

b[n_, i_, m_] := b[n, i, m] = Expand[If[n == 0, x, Sum[b[n - 1, j, Max[m, j]]*If[j > i, x, 1], {j, 1, m + 1}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1, 0]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, May 23 2016, after Alois P. Heinz *)

More terms from Franklin T. Adams-Watters, Jun 08 2006

Clarified definition and edited comment and example, Joerg Arndt, Mar 05 2016

A124102 a(n) = C(2n,n)*Bell(n).

Original entry on oeis.org

1, 2, 12, 100, 1050, 13104, 187572, 3009864, 53281800, 1028167140, 21427077100, 478684992240, 11394223609132, 287518731462200, 7658231740945200, 214521099763208400, 6299407928974197330, 193373975594104580880, 6189939300884798312700, 206159811915133359036600

Offset: 0

Zerinvary Lajos, Nov 26 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A000110, A056857, A056860.

with(combinat): a:=n->binomial(2*n,n)*bell(n): seq(a(n), n=0..20);

Table[Binomial[2n,n]*BellB[n],{n,0,20}] (* Harvey P. Dale, Apr 20 2012 *)

a(n) = A056857(2n+1,n+1) = A056860(2n+1,n+1). - Alois P. Heinz, May 25 2016

A259697 Triangle read by rows: T(n,k) = number of rook patterns on n X n board where bottom rook is in column k.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 9, 12, 15, 15, 1, 24, 32, 42, 52, 52, 1, 76, 99, 129, 166, 203, 203, 1, 279, 354, 451, 575, 726, 877, 877, 1, 1156, 1434, 1786, 2232, 2792, 3466, 4140, 4140, 1, 5296, 6451, 7883, 9664, 11881, 14621, 17884, 21147, 21147

Offset: 0

N. J. A. Sloane, Jul 05 2015

See Becker (1948/49) for precise definition.

This is the number of arrangements of non-attacking rooks on an n X n right triangular board where the bottom rook is in column k. The case of k=0 corresponds to the empty board where there is no bottom rook. - Andrew Howroyd, Jun 13 2017

			Triangle begins:
  1,
  1,  1,
  1,  2,  2,
  1,  4,  5,   5,
  1,  9, 12,  15,  15,
  1, 24, 32,  42,  52,  52,
  1, 76, 99, 129, 166, 203, 203,
  ...
From _Andrew Howroyd_, Jun 13 2017: (Start)
For n=3, the four solutions with the bottom rook in column 1 are:
  .      .      .      x
  . .    . x    x .    . .
  x . .  x . .  . . .  . . .
For n=3, the five solutions with the bottom rook in column 2 are:
  .      .      x      .       x
  . .    x .    . .    . x     . x
  . x .  . x .  . x .  . . .   . . .
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1274
H. W. Becker, Rooks and rhymes, Math. Mag. 22 (1948/49), 23-26. See Table V.
H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. [Annotated scanned copy]

Right edge is A000110.
Column k=1 is A005001.
Row sums are A000110(n+1).
Cf. A011971, A259691.

a11971[n_, k_] := Sum[Binomial[k, i]*BellB[n - k + i], {i, 0, k}];
T[, 0] = 1; T[n, k_] := Sum[a11971[i - 1, k - 1], {i, k, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

A(N)={my(T=matrix(N,N),U=matrix(N,N));T[1,1]=1;U[1,1]=1;
for(n=2,N,for(k=1,n, T[n,k]=if(k==1,T[n-1,n-1],T[n-1,k-1]+T[n,k-1]); U[n,k]=T[n,k]+U[n-1,k]));U}
{my(T=A(10));for(n=0,length(T),for(k=0,n,print1(if(k==0,1,T[n,k]),", "));print)} \\ Andrew Howroyd, Jun 13 2017

from sympy import bell, binomial
def a011971(n, k): return sum([binomial(k, i)*bell(n - k + i) for i in range(k + 1)])
def T(n, k): return 1 if k==0 else sum([a011971(i - 1, k - 1) for i in range(k, n + 1)])
for n in range(10): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 17 2017

T(n,0) = 1, T(n,k) = Sum_{i=k..n} A011971(i-1,k-1) for k>0. - Andrew Howroyd, Jun 13 2017

Terms a(28) and beyond from Andrew Howroyd, Jun 13 2017

A056861 Triangle T(n,k) is the number of restricted growth strings (RGS) of set partitions of {1..n} that have an increase at index k (1<=k

Original entry on oeis.org

1, 3, 2, 10, 7, 6, 37, 27, 23, 21, 151, 114, 97, 88, 83, 674, 523, 446, 403, 378, 363, 3263, 2589, 2217, 1999, 1867, 1785, 1733, 17007, 13744, 11829, 10658, 9923, 9452, 9145, 8942, 94828, 77821, 67340, 60689, 56380, 53541, 51644, 50361, 49484, 562595

Offset: 2

Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

Number of rises s_{k+1} > s_k in an RGS [s_1, ..., s_n] of a set partition of {1, ..., n}, where s_i is the subset containing i, and s_i <= 1 + max(j

Note that the number of equalities at any index is B(n-1), where B(n) are the Bell numbers. - Franklin T. Adams-Watters, Jun 08 2006

			For example, [1, 2, 1, 2, 2, 3] is the RGS of a set partition of {1, 2, 3, 4, 5, 6} and has 3 rises, at i = 1, i = 3 and i = 5.
1;
3,2;
10,7,6;
37,27,23,21;
151,114,97,88,83;
674,523,446,403,378,363;
3263,2589,2217,1999,1867,1785,1733;
17007,13744,11829,10658,9923,9452,9145,8942;
94828,77821,67340,60689,56380,53541,51644,50361,49484;
562595,467767,406953,367101,340551,322619,310365,301905,296011,291871;
3535027,2972432,2599493,2348182,2176575,2058068,1975425,1917290,1876075, 1846648,1825501;

W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [apparently unpublished, Joerg Arndt, Mar 05 2016]

Alois P. Heinz, Rows n = 2..100, flattened

Cf. Bell numbers A005493, A011965.

Cf. A056857-A056863.

b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[jJean-François Alcover, May 23 2016, after Alois P. Heinz *)

Edited and extended by Franklin T. Adams-Watters, Jun 08 2006
Clarified definition and edited comment and example, Joerg Arndt, Mar 08 2016
Several terms corrected, R. J. Mathar, Mar 08 2016

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cp's OEIS Frontend

A056860 Triangle T(n,k) = number of element-subset partitions of {1..n} with n-k+1 equalities (n >= 1, 1 <= k <= n).

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A259691 Triangle read by rows: T(n,k) number of arrangements of non-attacking rooks on an n X n right triangular board where the top rook is in row k (n >= 0, 1 <= k <= n+1).

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A056863 Related to triangle A056858 of number of rises in set partitions of n at a given index i.

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A124427 Sum of the sizes of the blocks containing the element 1 in all set partitions of {1,2,...,n}.

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A297926 Number of set partitions of [2n] in which the size of the first block is n.

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A175757 Triangular array read by rows: T(n,k) is the number of blocks of size k in all set partitions of {1,2,...,n}.

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A056858 Triangle of number of rises in restricted growth strings (RGS) for the set partitions of n.

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