A105486 -id:A105486 - cp's OEIS frontend

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.

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

A105490 Number of partitions of {1...n} containing 4 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 four 2-strings.

Original entry on oeis.org

5, 75, 780, 7105, 61390, 521640, 4440870, 38271750, 335892150, 3012721855, 27672081437, 260577574530, 2516984551900, 24942738309860, 253566501600240, 2643729700672284, 28259635983501165, 309569087038701420

Offset: 8

Augustine O. Munagi, Apr 10 2005

Number of partitions enumerated by A105481 in which the maximal length of consecutive integers in a block is 2.

With offset 4t, number of partitions of {1...N} containing 4 detached strings of t consecutive integers, where N = n + 4j, 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 four t-strings.

			a(8) = 5 because the partitions of {1,...,8} with 4 detached pairs of consecutive integers are 1256/3478, 1256/34/78, 12/3478/56, 1278/34/56, 12/34/56/78.

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

Cf. A105481, A105489, A105491, A105486.

seq(binomial(n-4, 4)*combinat[bell](n-5), n=8..28);
with(combinat): a:=n->sum(numbcomb(n-5, 3)*bell(n-5)/4, j=0..n-5): seq(a(n), n=8..28); # Zerinvary Lajos, Apr 25 2007

a(n) = binomial(n-4, 4)*Bell(n-5), which is the case r=4 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)*Bell(n-r*(t-1)-1).

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

A105494 Number of partitions of {1,...,n} containing 4 strings of 3 consecutive integers such that only v-strings of consecutive integers can appear in a block, where v = 1,2,3.

Original entry on oeis.org

5, 75, 855, 8665, 83485, 788515, 7424515, 70378930, 675685240, 6594991405, 65598204272

Offset: 12

Augustine O. Munagi, Apr 11 2005

Partitions enumerated by A105486 in which the maximal length of consecutive integers in a block is 3.

			a(12)=5, the enumerated partitions are (1,2,3,7,8,9)(4,5,6,10,11,12),
(1,2,3,7,8,9)(4,5,6)(10,11,12), (1,2,3)(4,5,6,10,11,12)(7,8,9),
(1,2,3,10,11,12)(4,5,6)(7,8,9), (1,2,3)(4,5,6)(7,8,9) (10,11,12).

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. A105486, A105490, A105493.

a(n)=Sum(w(n, k, 4), k=1...n), where w(n, k, 4) is the case r=4 of w(n, k, r) given by w(m, k, r)=w(m-1, k-1, r)+(k-1)w(m-1, k, r)+w(m-2, k-1, r)+(k-1)w(m-2, k, r) +w(m-3, k-1, r-1)+(k-1)w(m-3, k, r-1) r=0, 1, ..., floor(n/3), k=1, 2, ..., n-2r, w(n, k, 0)=sum(binomial(n-j, j)*S2(n-j-1, k-1), j=0..floor(n/2)).

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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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A105490 Number of partitions of {1...n} containing 4 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 four 2-strings.

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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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A105494 Number of partitions of {1,...,n} containing 4 strings of 3 consecutive integers such that only v-strings of consecutive integers can appear in a block, where v = 1,2,3.

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