A105487 -id:A105487 - cp's OEIS frontend

A105482 Number of partitions of {1...n} containing 5 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, 6, 42, 280, 1890, 13104, 93786, 694584, 5328180, 42336294, 348272925, 2963993760, 26073738236, 236857536216, 2219777316216, 21441389281680, 213260412549303, 2182163481418536, 22951202450444191, 247914874683742728

Offset: 6

Augustine O. Munagi, Apr 10 2005

			a(7) = 6 because the partitions of {1,2,3,4,5,6,7} with 5 pairs of consecutive integers are 123456/7,12345/67,1234/567,123/4567,12/34567,1/234567.

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

Cf. A105481, A105487, A105491.

seq(binomial(n-1,5)*combinat[bell](n-6),n=6..26);

a(n) = binomial(n-1, 5)*Bell(n-6), the case r = 5 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>=5, a(n+1)=(-1)^(n-5)*coeff(charpoly(A,x),x^5). [Milan Janjic, Jul 08 2010]
E.g.f.: (1/5!) * Integral (x^5 * exp(exp(x) - 1)) dx. - Ilya Gutkovskiy, Jul 10 2020

A105486 Number of partitions of {1...n} containing 4 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, 11, 50, 255, 1362, 7746, 46556, 294965, 1963865, 13703812, 99974851, 760824922, 6027441398, 49616033975, 423649629415, 3746306203604, 34259548971914, 323564415957687, 3152120868598090, 31638011553779137, 326825518800658174, 3471291152755614386

Offset: 6

Augustine O. Munagi, Apr 10 2005

nonn

			a(7) = 2 because the partitions of {1,...,7} with 4 strings of 3 consecutive integers are 123456/7, 1/234567.

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

Cf. A105485, A105487, A105490, A105494.

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: A105486 := proc(n) local k ; add(c(n,k,4),k=1..n) ; end: for n from 6 to 29 do printf("%d, ",A105486(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]];
A105486[n_] := Sum[c[n, k, 4], {k, 1, n}];
Table[A105486[n], {n, 6, 29}] (* Jean-François Alcover, May 10 2023, after R. J. Mathar *)

a(n) = Sum_{k=1..n} (c(n, k, 4), ), where c(n, k, 4) is the case r=4 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

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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A105482 Number of partitions of {1...n} containing 5 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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A105486 Number of partitions of {1...n} containing 4 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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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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