cp's OEIS Frontend

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001

a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003

a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003

Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004

Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004

The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008

Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008

First column of A092582. - Mats Granvik, Feb 08 2009

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

For n>1: a(n) = A173333(n,2). - Reinhard Zumkeller, Feb 19 2010

Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010

For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011.

The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011

Also [1, 1] together with the row sums of triangle A162608. - Omar E. Pol, Mar 09 2012

a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11, n=3: 112, n=4: 1123, 1132, 1213, n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012

For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014

In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015

Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017

Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018

Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018

For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019

Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019

a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020

For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020

For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020

From Wolfdieter Lang, Jun 27 2012: (Start)

T(n-1,k), k=1,...,n-1, gives the number of representative necklaces with n beads (C_N symmetry) of n+1-k distinct colors, say c[1],c[2],...,c[n-k+1], corresponding to the color signature determined by the partition k,1^(n-k) of n. The representative necklaces have k beads of color c[1]. E.g., n=4, k=2: partition 2,1,1, color signature (parts as exponents) c[1]c[1]c[2]c[3], 3=T(3,2) necklaces (write j for color c[j]): cyclic(1123), cyclic(1132) and cyclic(1213). See A212359 for the numbers for general partitions or color signatures. (End)

The row lengths sequence is A000041(n), n >= 1.

The partitions are ordered like in Abramowitz-Stegun (A-St order). For the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used.

A bracelet with n beads (n-bracelet) has the dihedral D_n symmetry group of degree n (order 2n). In addition to cyclic C_n operations, also a turnover (in 3-space) or a reflection (in 2-space) is allowed. In the Harary-Palmer reference, p. 44, the term necklace is used instead of bracelet.

a(n,k) gives the number of representative n-bracelets, with up to n colors for each bead, belonging to the k-th partition of n in A-St order in the following way. Write this partition with nonincreasing parts (this is the reverse of the partition as given by A-St), e.g., [3,1^2], not [1^2,3], which is written as [3,1,1], a partition of n=5. In general a (reversed) partition of n is written as [p[1],p[2],...,p[m]], with p[1] >= p[2] >= ... >= p[m] >= 1, with m the number of parts. To each such partition of n corresponds an n-multiset obtained by 'exponentiation'. For more details see the W. Lang link in A213938 with more details as well as a list of multiset signatures and corresponding multiset representatives. For the given example the 5-multiset is {1^3,2^1,3^1}={1,1,1,2,3}. In general, {1^p[1],2^p[2],...,m^p[m]}. We will also use a list notation with square brackets for these multisets. Such an n-multiset representative (of a repetition class defined by the exponents, also called signature) encodes the representative n-bracelet color monomial by c[1]^p[1]*c[2]^p[2]*...*c[m]^p[m]. For the example one has c[1]^3*c[2]*c[3]. The number of 5-bracelets with this color assignment is a(5,4) because [3,1,1] is the 4th partition of 5 in A-St order. The a(5,4)=2 non-equivalent 5-bracelets with this color assignment are cyclic(c[1]c[1]c[1]c[2]c[3]) and cyclic(c[1]c[1]c[2]c[1]c[3]). For the necklace case c[1]c[1]c[1]c[3]c[2] and c[1]c[1]c[3]c[1]c[2] (both taken cyclically) also have to be counted, but due to a turn over (or a reflection) they become equivalent to the two given bracelets, respectively.

Such a set of a(n,k) n-bracelets for the given color signature stands for other sets of the same order when different colors from the repertoire {c[1],...,c[n]} are chosen. In the example, the partition [3,1,1] with the representative multiset [1^3,2,3] stands for all-together 5*binomial(4,2) = 30 such sets, each leading to 2 possible non-equivalent 5-bracelet arrangements. Thus one has all-together 30*2=60 5-bracelets with color signature determined from the partition [3,1,1]. See the partition array A213941 for these total bracelet numbers.

a(n,k) is computed from the cycle index Z(D_n) for the dihedral group (see A212355 and the link given there) after the variables x_j have been replaced by the j-th power sum sum(c[i]^j,i=1..n), abbreviated as Z(D_n,c_n) with c_n:=sum(c[i],i=1..n), n >= 1. The coefficient of the representative color multinomial determined by the k-th partition of n in A-St order, as explained above, is a(n,k). See the Harary-Palmer reference, p. 36, Theorem (PET) with A = D_n and p. 37 eq. (2.2.11) for the cycle index polynomial Z(D_n). See the W. Lang link for more details.

The row sums are given by A213943.

This triangle is obtained from the partition array A213939 by summing in row n, for n>=1, all entries related to partitions of n with the same number of parts m.

a(n,m) is the number of bracelets of n beads (dihedral D_n symmetry) corresponding to the representative color multinomials obtained from all partitions of n with m parts by 'exponentiation', hence only m from the available n colors are present. As a representative multinomial of each of the p(n,m)=A008284(n,m) such m-color classes we take the one where the considered m part partition of n, [p[1],...,p[m]], written in nonincreasing order, is distributed as exponents on the color indices like c[1]^p[1]*...*c[m]^p[m]. That is only the first m colors from the n available ones are involved.

See the comments on A212359 for the Abramowitz-Stegun (A-St) order of partitions, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions.

The row sums of this triangle coincide with the ones of array A213939, and they are given by A213943.

Number of n-length bracelets w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226874). - Andrew Howroyd, Sep 26 2017

This array is obtained by multiplying the entry of the array A213939(n,k) (number of bracelets (dihedral D_n symmetry) with n beads, each available in n colors, with color representative given by the n-multiset representative obtained from the k-th partition of n in A-St order after 'exponentiation') with the entry of the array A035206(n,k) (number of members in the equivalence class represented by the color multiset considered for A213939(n,k)): a(n,k)=A213939(n,k)*A035206(n,k), k=1..p(n)=A000041(n), n>=1. The row sums then give the total number of bracelets with n beads from n colors, given by A081721(n).

See A212359 for references, the 'exponentiation', and a link. For multiset signatures and representative multisets defining color multinomials see also a link in A213938.

The corresponding triangle with the summed row entries related to partitions of n with fixed number of parts is A214306.

a(n-1), n>=5, gives the number of necklaces with n beads (C_n symmetry) with color signature determined from the partition 2^2,1^(n-4) of n. Only n-2 distinct colors, say c[1], c[2], ..., c[n-2] are used, and the representative necklaces have the color c[1] and c[2] each twice. E.g., n=5, partition 2,2,1, color signature (take the parts as exponents) c[1]c[1]c[2]c[2]c[3], with the a(4)=6 necklaces (write j for color c[j]) 11223, 11232, 11322, 12213, 12123 and 12132, all taken cyclically. See A212359 for the numbers for general partitions or color signatures. - Wolfdieter Lang, Jun 27 2012

This triangle is obtained from the array A213941 by summing in row n, for n >= 1, all entries related to partitions of n with the same number of parts m.

a(n,m) is the total number of necklaces of n beads (dihedral D_n symmetry) corresponding to all the color multinomials obtained from all p(n,m) = A008284(n,m) partitions of n with m parts, written in nonincreasing form, by 'exponentiation'. Therefore only m from the available n colors are present, and a(n,m) gives the number of bracelets with n beads with only m of the n available colors present, for m from 1,2,...,n, and n >= 1. All of the possible color assignments are counted.

See the comments on A212359 for the Abramowitz-Stegun (A-St) order of partitions, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions. See a link in A213938 for representative multisets for given signature used to define color multinomials.

The row sums of this triangle coincide with the ones of array A213941, and they are given by A081721.

This table coincides with A173333 but has an extra main diagonal with entries 1.

a(n,k) is the number of necklaces of n beads (C_N symmetry), with colors from the repertoire {c[1],c[2],...,c[n]}, corresponding to the representative color multinomials obtained from the partition [k,1^(n-k)] of n with m=n-k+1 parts by 'exponentiation' (taking the parts in the given order as exponents of the colors), hence only m from the available n colors are present. As representative necklaces one takes the ones where the color c[1] appears k times. In particular, for k=1 the partition is [1^n] and all n colors are used, and there are (n-1)! necklaces from permuting the n colors.

a(n,k) appears in the representative necklace partition array A212359 in row n at the position l(n,n+1-k,1), with l(n,m,1) the position of the first partition with m parts in the list of partitions of n in A-St order. E.g., n=5, k=4: l(5,5-3,1) =2 with the partition [4,1] (used in reverse order compared to A-St).

See the comments on A212359 for the Abramowitz-Stegun (A-St) reference, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions.

The row sums of this triangle are given by A213937.

This array is obtained by multiplying the entry of the array A212359(n,k) (number of necklaces (C_n symmetry) with n beads, each available in n colors, with color representative given by the n-multiset representative obtained from the k-th partition of n in A-St order after 'exponentiation') with the entry of the array A035206(n,k) (number of members in the equivalence class represented by the color multiset considered for A212359(n,k)): a(n,k)=A212359(n,k)* A035206(n,k), k=1..p(n)= A000041(n), n>=1. The row sums then give the total number of necklaces with beads from n colors, given by A056665(n).

See A212359 for references, the 'exponentiation', and a link.

The corresponding triangle with the summed row entries which belong to partitions of n with fixed number of parts is A213935. [From Wolfdieter Lang, Jul 12 2012]

This triangle is obtained from the partition array A212359 by summing in the row number n, for n>=1, all entries related to partitions of n with the same number of parts m.

a(n,m) is the number of necklaces of n beads (C_N symmetry) corresponding to the representative color multinomials obtained from all partitions of n with m parts by 'exponentiation', hence only m from the available n colors are present. As a representative multinomial of each of the p(n,m)=A008284(n,m) such m-color classes we take the one where the considered m part partition of n, [p[1],...,p[m]], written in a nonincreasing way, is distributed as exponents over c[1]^p[1]*...*c[m]^p[m]. That is only the first m colors from the n available ones are involved.

See the comments on A212359 for the Abramowitz-Stegun (A-St) order of partitions, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions.

The row sums of this triangle coincide with the ones of array A212359, and they are given by A072605.

Number of necklaces with n beads w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w (necklace analog of A226874). - Andrew Howroyd, Dec 20 2017