Augustine O. Munagi - cp's OEIS frontend

A330774 Number of n-color perfect compositions of n.

1, 1, 1, 5, 1, 11, 1, 61, 7, 15, 1, 259, 1, 19, 17, 1901, 1, 383, 1, 511, 21, 27, 1, 14147, 11, 31, 187, 859, 1, 1403, 1, 147661, 29, 39, 25, 39307, 1, 43, 33, 42351, 1, 2303, 1, 1843, 947, 51, 1, 1815811, 15, 1255, 41, 2479, 1, 46697, 33, 97339, 45, 63, 1, 219347, 1

Offset: 0

Augustine O. Munagi, Dec 30 2019

nonn

An n-color perfect composition of v is a composition into j types of each part j whose sequence of parts contains one composition of every positive integer less than n.

			a(5)=11 because v=5 has eleven n-color perfect compositions: (1,1,1,1,1), (1,2,2),(2,2,1), (1,2',2'), (2',2',1), (1,1,3), (3,1,1), (1,1,3'), (3',1,1), (1,1,3''), (3'',1,1).

A. O. Munagi, Perfect Compositions of Numbers, J. Integer Seq. 23 (2020), art. 20.5.1.

Cf. A330773, A165552, A001906, A002033, A074206.

A330773 Number of perfect compositions of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 11, 3, 5, 1, 27, 1, 5, 5, 49, 1, 27, 1, 27, 5, 5, 1, 163, 3, 5, 11, 27, 1, 49, 1, 261, 5, 5, 5, 231, 1, 5, 5, 163, 1, 49, 1, 27, 27, 5, 1, 1109, 3, 27, 5, 27, 1, 163, 5, 163, 5, 5, 1, 435, 1, 5, 27, 1631, 5, 49, 1, 27, 5, 49, 1, 2055, 1, 5, 27, 27, 5, 49, 1

Offset: 0

Augustine O. Munagi, Dec 30 2019

A perfect composition of n is one whose sequence of parts contains one composition of every positive integer less than n.

			a(7) = 11 because the perfect compositions are 1111111, 1222, 2221, 1114, 4111, 124, 142, 214, 241, 412, 421.
For example, 241 generates the compositions of 1,...,6: 1,2,21,4,41,24.

Alois P. Heinz, Table of n, a(n) for n = 0..20000
A. O. Munagi, Perfect Compositions of Numbers, J. Integer Seq. 23 (2020), art. 20.5.1.

Cf. A001222, A002033, A074206, A251683, A330774.

b:= proc(n) option remember; expand(x*(1+add(b(n/d),
       d=numtheory[divisors](n) minus {1, n})))
    end:
a:= n-> (p-> add(coeff(p, x, i)*i!, i=1..degree(p)))(b(n+1)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 15 2020

b[n_] := b[n] = x(1+Sum[b[n/d], {d, Divisors[n]~Complement~{1, n}}]);
a[n_] := With[{p = b[n+1]}, Sum[Coefficient[p, x, i] i!, {i, Exponent[p, x]}]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

a(1)=1, a(n) = Sum_{k=1..Omega(n+1)} k! * A251683(n+1,k), n>1.

A160086 a(n) = A104725(n) - A074206(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 7, 0, 3, 0, 3, 0, 0, 0, 25, 0, 0, 1, 3, 0, 6, 0, 36, 0, 0, 0, 36, 0, 0, 0, 25, 0, 6, 0, 3, 3, 0, 0, 152, 0, 3, 0, 3, 0, 25, 0, 25, 0, 0, 0, 69, 0, 0, 3, 171, 0, 6, 0, 3, 0, 6, 0, 279, 0, 0, 3, 3, 0, 6, 0, 152, 7, 0, 0, 69, 0, 0, 0, 25, 0, 69, 0, 3, 0, 0, 0

Offset: 0

Augustine O. Munagi, May 01 2009

nonn

a(n) is also the excess of the number of labeled factorizations of n over the number of ordered factorizations (see the Munagi link for definition of labeled factorization)

			a(8)=1 because A074206(8)=4 and A104725(8)=5, so a(8)=5-4. The only labeled factorization of 8 which is not an ordered factorization is (2_1.2_3)(2_2). a(9)=0 because A074206(9)=2=A104725(9). The labeled factorizations of 9, namely (9_1) and (3_1)(3_2), are also ordered factorizations.

Antti Karttunen, Table of n, a(n) for n = 0..10000 (computed from the b-files of A074206 and A104725)
A. O. Munagi, Labeled factorization of integers, Electron. J. Combin. 16 (2009), no. 1, Research Paper 50, 17pp.

Cf. A074206, A104725.

a:=proc(n::integer) local u, r, i, j, k; if n<2 then return 0; end if; u:=map(x->x[2], ifactors(n)[2]); r:=add(u[i], i=1..nops(u)); add(add((-1)^i*binomial(k, i)*product(binomial(u[j]+k-i-1, u[j]), j=1..nops(u)), i=0..k-1)*(bell(k-1)-1), k=1..r); end proc: seq(a(n),n=0..99);

a(n) = Sum(ordfac(n,k)*(Bell(k-1)-1),k=1..Omega(n)), where ordfac(n,k)=number of ordered factorizations of n into k factors.

A160085 Number of ordered complementing systems of subsets of {0, 1, ..., n-1} (see A104725).

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 5, 1, 13, 3, 5, 1, 33, 1, 5, 5, 75, 1, 33, 1, 33, 5, 5, 1, 261, 3, 5, 13, 33, 1, 61, 1, 541, 5, 5, 5, 375, 1, 5, 5, 261, 1, 61, 1, 33, 33, 5, 1, 2405, 3, 33, 5, 33, 1, 261, 5, 261, 5, 5, 1, 717, 1, 5, 33, 4683, 5, 61, 1, 33, 5, 61, 1, 4549, 1, 5, 33, 33, 5, 61, 1, 2405

Offset: 0

Augustine O. Munagi, May 01 2009

nonn

Also number of permuted labeled factorizations of n (see the Munagi link for definition and examples)

			a(6) = 5: there are 3 complementing systems of subsets of {0,1,2,3,4,5} namely {{0,1,2,3,4,5}}, {{0,1,2},{0,3}} and {{0,1},{0,2,4}} (see A104725). Permuting the components gives 2 additional systems: {{0,3},{0,1,2}} and {{0,2,4},{0,1}}. Thus since {{0,1,2},{0,3}} is a complementing system of subsets of {0,1,2,3,4,5} we have 0 = 0 + 0, 1 = 1 + 0, 2 = 2 + 0, 3 = 0 + 3, 4 = 1 + 3, 5 = 2 + 3.

A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.
A. O. Munagi, Labeled factorization of integers, Electron. J. Combin. 16 (2009), no. 1, Research Paper 50, 17pp.

Cf. A104725.

a:=proc(n::integer) local u, r, i, j, k; if n<1 then return 0; elif n=1 then return 1; end if; u:=map(x->x[2], ifactors(n)[2]); r:=add(u[i], i=1..nops(u)); add(add(k!*add((-1)^i*binomial(t, i)*product(binomial(u[j]+t-i-1, u[j]), j=1..nops(u)), i=0..t-1)*stirling2(t-1,k-1), t=k..r),k=1..r); end proc: seq(a(n),n=0..99);

a(0) = 0, a(1) = 1, a(n) = Sum_{j=1..A001222(n)} (Sum_{k=1..j} k!Stirling2(j-1,k-1)), n > 1, where ordfac(n,k) = number of ordered factorizations of n into k factors.

A131456 Number of q-partial fraction summands of the reciprocal of n-th cyclotomic polynomial.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 7, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 10, 1, 2, 1, 2, 1, 2, 1, 2, 7

Offset: 1

Augustine O. Munagi, Jul 12 2007

nonn

Let Phi(n,q) be the n-th cyclotomic polynomial in q. The q-partial fraction decomposition of 1/Phi(n,q) is a representation of 1/Phi(n,q) as a finite sum of functions v(q)/(1-q^m)^t, such that m<=n and degree(v)A000010).

			(i) a(3)=1 because 1/Phi(3,q)=(1-q)/(1-q^3);
(ii) a(6)=2 because 1/Phi(6,q)=(-1-q)/(1-q^3) + (2+2q)/(1-q^6).

Augustine O. Munagi, Computation of q-partial fractions, INTEGERS: Electronic Journal Of Combinatorial Number Theory, 7 (2007), #A25.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A051664 (Number of terms in n-th cyclotomic polynomial).

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).

A105492 Number of partitions of {1,...,n} containing 2 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

1, 6, 36, 210, 1260, 7833, 50701, 342126, 2406645, 17633820, 134427468, 1064801442, 8751834839, 74540800014

Offset: 6

Augustine O. Munagi, Apr 11 2005

Partitions enumerated by A105484 in which the maximal length of consecutive integers in a block is 3.

			a(7)=6; the enumerated partitions are 123567/4, 1237/456, 1567/234, 123/456/7, 123/4/567, 1/234/567.

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. A105484, A105488, A105493.

a(n)=Sum(w(n, k, 2), k=1...n), where w(n, k, 2) is the case r=2 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)).

A105493 Number of partitions of {1,...,n} containing 3 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

2, 20, 170, 1340, 10375, 80652, 636990, 5143740, 42613980, 362863600, 3178544754, 28650249848

Offset: 9

Augustine O. Munagi, Apr 11 2005

Partitions enumerated by A105485 in which the maximal length of consecutive integers in a block is 3.

			a(9)=2, the enumerated partitions are 123/789/456, 123/456/789.

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. A105485, A105489, A105492.

a(n)=Sum(w(n, k, 3), k=1...n), where w(n, k, 3) is the case r=3 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)).

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)).

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).

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User: Augustine O. Munagi

A330774 Number of n-color perfect compositions of n.

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A330773 Number of perfect compositions of n.

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A160086 a(n) = A104725(n) - A074206(n).

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A160085 Number of ordered complementing systems of subsets of {0, 1, ..., n-1} (see A104725).

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A131456 Number of q-partial fraction summands of the reciprocal of n-th cyclotomic polynomial.

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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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A105492 Number of partitions of {1,...,n} containing 2 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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A105493 Number of partitions of {1,...,n} containing 3 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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