A105482 -id:A105482 - cp's OEIS frontend

A105485 Number of partitions of {1...n} containing 3 strings of 3 consecutive integers, where each string is counted within a block and a string of more than 3 consecutive integers are counted three at a time.

1, 2, 10, 44, 215, 1112, 6141, 35968, 222659, 1451770, 9939702, 71265036, 533744979, 4166533826, 33831424388, 285213338300, 2492259168784, 22538314947452, 210639529104328, 2031804667766532, 20203377516199587, 206861906112012524, 2178715175981722659

Offset: 5

Augustine O. Munagi, Apr 10 2005

nonn

			a(6)=2 because the partitions of {1,...,6} with 3 strings of 3 consecutive integers are 12345/6, 1/23456.

Augustine O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc., 2005:3 (2005), 451-463.

Cf. A105482, A105484, A105489, A105493.

c := proc(n,k,r) option remember ; local j ; if r =0 then add(binomial(n-j,j)*combinat[stirling2](n-j-1,k-1),j=0..floor(n/2)) ; else if r <0 or r > n-k-1 then RETURN(0) fi ; if n <1 then RETURN(0) fi ; if k <1 then RETURN(0) fi ; RETURN( c(n-1,k-1,r)+(k-1)*c(n-1,k,r)+c(n-2,k-1,r)+(k-1)*c(n-2,k,r) +c(n-1,k,r-1)-c(n-2,k-1,r-1)-(k-1)*c(n-2,k,r-1) ) ; fi ; end: A105485 := proc(n) local k ; add(c(n,k,3),k=1..n) ; end: for n from 5 to 28 do printf("%d, ",A105485(n)) ; od ; # R. J. Mathar, Feb 20 2007

S2[_, -1] = 0;
S2[n_, k_] = StirlingS2[n, k];
c[n_, k_, r_] := c[n, k, r] = Which[r == 0, Sum[Binomial[n - j, j]*S2[n - j - 1, k - 1], {j, 0, Floor[n/2]}], r < 0 || r > n - k - 1, 0, n < 1, 0, k < 1, 0, True, c[n - 1, k - 1, r] + (k - 1)*c[n - 1, k, r] + c[n - 2, k - 1, r] + (k - 1)*c[n - 2, k, r] + c[n - 1, k, r - 1] - c[n - 2, k - 1, r - 1] - (k - 1)*c[n - 2, k, r - 1]];
A105485[n_] := Sum[c[n, k, 3], {k, 1, n}];
Table[A105485[n], {n, 5, 28}] (* Jean-François Alcover, May 10 2023, after R. J. Mathar *)

a(n) = Sum_{k=1..n} c(n, k, 3), where c(n, k, 3) is the case r=3 of c(n, k, r) given by c(n, k, r)=c(n-1, k-1, r)+(k-1)c(n-1, k, r)+c(n-2, k-1, r)+(k-1)c(n-2, k, r)+c(n-1, k, r-1)-c(n-2, k-1, r-1)-(k-1)c(n-2, k, r-1), r=0, 1, .., n-k-1, k=1, 2, .., n-2r, c(n, k, 0) = Sum_{j= 0..floor(n/2)} binomial(n-j, j)*S2(n-j-1, k-1).

More terms from R. J. Mathar, Feb 20 2007

A105481 Number of partitions of {1...n} containing 4 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time.

Original entry on oeis.org

1, 5, 30, 175, 1050, 6552, 42630, 289410, 2049300, 15120105, 116090975, 926248050, 7668746540, 65793760060, 584151925320, 5360347320420, 50776288702215, 495946245776940, 4989391837053085, 51648932225779735, 549620905409062872

Offset: 5

Augustine O. Munagi, Apr 10 2005

			a(6) = 5 because the partitions of {1,2,3,4,5,6} with 4 pairs of consecutive integers are 12345/6,1234/56,123/456,12/3456,1/23456.

A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.

A. O. Munagi, Set Partitions with Successions and Separations, IJMMS 2005:3 (2005), 451-463.

Cf. A105480, A105482, A105486, A105491, A105494.

seq(binomial(n-1,4)*combinat[bell](n-5),n=5..25);

a(n) = binomial(n-1, 4)*Bell(n-5), the case r = 4 in the general case of r pairs: c(n, r) = binomial(n-1, r)*B(n-r-1).
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=4, a(n+1)=(-1)^(n-4)*coeff(charpoly(A,x),x^4). [Milan Janjic, Jul 08 2010]
E.g.f.: (1/4!) * Integral (x^4 * exp(exp(x) - 1)) dx. - Ilya Gutkovskiy, Jul 10 2020

A105487 Number of partitions of {1...n} containing 5 strings of 3 consecutive integers, where each string is counted within a block and a string of more than 3 consecutive integers are counted three at a time.

Original entry on oeis.org

1, 2, 12, 56, 297, 1632, 9531, 58634, 379371, 2574254, 18276457, 135463074, 1046041114, 8399533370, 70013963418, 604840440328, 5407301690915, 49958478263502, 476403955991034, 4683463406478004, 47414166201239781, 493803423334040824, 5285548108715948453

Offset: 7

Augustine O. Munagi, Apr 10 2005

nonn

			a(8) = 2 because the partitions of {1,...,8} with 5 strings of 3 consecutive integers are 1234567/8, 1/2345678.

Augustine O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc., 2005:3 (2005), 451-463.

Cf. A105486, A105482.

c := proc(n,k,r) option remember ; local j ; if r =0 then add(binomial(n-j,j)*combinat[stirling2](n-j-1,k-1),j=0..floor(n/2)) ; else if r <0 or r > n-k-1 then RETURN(0) fi ; if n <1 then RETURN(0) fi ; if k <1 then RETURN(0) fi ; RETURN( c(n-1,k-1,r)+(k-1)*c(n-1,k,r)+c(n-2,k-1,r)+(k-1)*c(n-2,k,r) +c(n-1,k,r-1)-c(n-2,k-1,r-1)-(k-1)*c(n-2,k,r-1) ) ; fi ; end: A105487 := proc(n) local k ; add(c(n,k,5),k=1..n) ; end: for n from 7 to 30 do printf("%d, ",A105487(n)) ; od ; # R. J. Mathar, Feb 20 2007

S2[_, -1] = 0;
S2[n_, k_] = StirlingS2[n, k];
c[n_, k_, r_] := c[n, k, r] = Which[
   r == 0, Sum[Binomial[n - j, j]*S2[n - j - 1, k - 1],
      {j, 0, Floor[n/2]}],
   r < 0 || r > n - k - 1, 0,
   n < 1, 0,
   k < 1, 0,
   True, c[n - 1, k - 1, r] +
      (k - 1)*c[n - 1, k, r] +
      c[n - 2, k - 1, r] +
      (k - 1)*c[n - 2, k, r] +
      c[n - 1, k, r - 1] -
      c[n - 2, k - 1, r - 1] -
      (k - 1)*c[n - 2, k, r - 1]];
A105487[n_] := Sum[c[n, k, 5], {k, 1, n}];
Table[A105487[n], {n, 7, 30}] (* Jean-François Alcover, May 10 2023, after R. J. Mathar *)

a(n) = Sum_{k=1..n} c(n, k, 5), where c(n, k, 5) is the case r=5 of c(n, k, r) given by c(n, k, r)=c(n-1, k-1, r)+(k-1)c(n-1, k, r)+c(n-2, k-1, r)+(k-1)c(n-2, k, r)+c(n-1, k, r-1)-c(n-2, k-1, r-1)-(k-1)c(n-2, k, r-1), r=0, 1, .., n-k-1, k=1, 2, .., n-2r, c(n, k, 0) = Sum_{j= 0..floor(n/2)} binomial(n-j, j)*S2(n-j-1, k-1).

More terms from R. J. Mathar, Feb 20 2007

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

A105491 Number of partitions of {1...n} containing 5 detached pairs of consecutive integers, i.e., partitions in which only 1- or 2-strings of consecutive integers can appear in a block and there are exactly five 2-strings.

Original entry on oeis.org

15, 312, 4263, 49112, 521640, 5329044, 53580450, 537427440, 5422899339, 55344162874, 573270663966, 6040762924560, 64851119605636, 709986204480672, 7931189102016852, 90430835147203728, 1052534895931584828

Offset: 10

Augustine O. Munagi, Apr 10 2005

Number of partitions enumerated by A105482 in which the maximal length of consecutive integers in a block is 2.

With offset 5t, number of partitions of {1,...,N} containing 5 detached strings of t consecutive integers, where N=n+5j, t=2+j, j = 0,1,2,..., i.e., partitions of {1,...,N} in which only v-strings of consecutive integers can appear in a block, where v=1 or v=t and there are exactly five t-strings.

			a(10)=15; the enumerated 15 partitions of {1,...,10} with 5 detached pairs of consecutive integers include (1,2,5,6,9,10)(3,4,7,8) and (1,2,9,10)(3,4,7,8)(5,6).

A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.

A. O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005),451-463.

Cf. A105482, A105490, A105487.

seq(binomial(n-5,5)*combinat[bell](n-6),n=10..30);

a(n)=binomial(n-5, 5)*Bell(n-6), which is the case r=5 in the general case of r pairs, d(n, r)=binomial(n-r, r)*Bell(n-r-1), which is the case t=2 of the general formula d(n, r, t)=binomial(n-r*(t-1), r)*B(n-r*(t-1)-1).

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A105485 Number of partitions of {1...n} containing 3 strings of 3 consecutive integers, where each string is counted within a block and a string of more than 3 consecutive integers are counted three at a time.

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A105481 Number of partitions of {1...n} containing 4 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time.

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A105487 Number of partitions of {1...n} containing 5 strings of 3 consecutive integers, where each string is counted within a block and a string of more than 3 consecutive integers are counted three at a time.

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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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A105491 Number of partitions of {1...n} containing 5 detached pairs of consecutive integers, i.e., partitions in which only 1- or 2-strings of consecutive integers can appear in a block and there are exactly five 2-strings.

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