A105490 -id:A105490 - cp's OEIS frontend

A105480 Number of partitions of {1...n} containing 3 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, 4, 20, 100, 525, 2912, 17052, 105240, 683100, 4652340, 33168850, 246999480, 1917186635, 15480884720, 129811538960, 1128494172720, 10155257740443, 94465951576560, 907162152191470, 8982422995787780, 91603484234843812

Offset: 4

Augustine O. Munagi, Apr 10 2005

			a(5) = 4 because the partitions of {1,2,3,4,5} with 3 pairs of consecutive integers are 1234/5,123/45,12/345,1/2345.

A. O. Munagi, Set partitions with successions and separations, Int. J. Math. Math. Sci. (IJMMS) vol 2005 no 3 (2005) pp 451-463.

Cf. A105479, A105481, A105485, A105490.

seq(binomial(n-1,3)*combinat[bell](n-4),n=4..25);

a(n) = binomial(n-1, 3)*Bell(n-4), the case r = 3 in the general case of r pairs: c(n, r) = binomial(n-1, r)*B(n-r-1).
O.g.f. for c(n,r) is exp(-1)*Sum(x^(r+1)/(n!*(1-n*x)^(r+1)),n=0..infinity). - Vladeta Jovovic, Feb 05 2008
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>=3, a(n+1)=(-1)^(n-3)*coeff(charpoly(A,x),x^3). [Milan Janjic, Jul 08 2010]
E.g.f.: (1/3!) * Integral (x^3 * exp(exp(x) - 1)) dx. - Ilya Gutkovskiy, Jul 10 2020

A105489 Number of partitions of {1...n} containing 3 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 three 2-strings.

Original entry on oeis.org

2, 20, 150, 1040, 7105, 49112, 347760, 2537640, 19135875, 149285400, 1205088742, 10062575068, 86859191510, 774456785200, 7126496659960, 67617733760064, 660932425168071, 6649326113764980, 68793130453044760, 731299516881396540

Offset: 6

Augustine O. Munagi, Apr 10 2005

Number of partitions enumerated by A105480 in which the maximal length of consecutive integers in a block is 2.

With offset 3t, number of partitions of {1...N} containing 3 detached strings of t consecutive integers, where N = n + 3j, 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 three t-strings.

			a(6) = 2 because the partitions of {1,2,3,4,5,6} with 3 detached pairs of consecutive integers are 12/34/56, 1256/34.

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

Cf. A105480, A105485, A105488, A105490.

seq(binomial(n-3,3)*combinat[bell](n-4),n=6..25);
a:=n->sum(numbcomb (n,2)*bell(n)/3, j=0..n): seq(a(n), n=2..21); # Zerinvary Lajos, Apr 25 2007

a(n) = binomial(n-3, 3)*Bell(n-4), which is the case r=3 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).

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).

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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A105480 Number of partitions of {1...n} containing 3 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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A105489 Number of partitions of {1...n} containing 3 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 three 2-strings.

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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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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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