Displaying 1-10 of 15 results found.
Number of partitions of n-set with distinct block sizes.
+10
102
1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847
COMMENTS
Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022
LINKS
Knopfmacher, A., Odlyzko, A. M., Pittel, B., Richmond, L. B., Stark, D., Szekeres, G. and Wormald, N. C., The asymptotic number of set partitions with unequal block sizes, Electron. J. Combin., 6 (1999), no. 1, Research Paper 2, 36 pp.
FORMULA
E.g.f.: Product_{m >= 1} (1+x^m/m!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018
EXAMPLE
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
{{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}}
{{1},{2,3}} {{1},{2,3,4}} {{1},{2,3,4,5}}
{{1,2},{3}} {{1,2,3},{4}} {{1,2},{3,4,5}}
{{1,3},{2}} {{1,2,4},{3}} {{1,2,3},{4,5}}
{{1,3,4},{2}} {{1,2,3,4},{5}}
{{1,2,3,5},{4}}
{{1,2,4},{3,5}}
{{1,2,4,5},{3}}
{{1,2,5},{3,4}}
{{1,3},{2,4,5}}
{{1,3,4},{2,5}}
{{1,3,4,5},{2}}
{{1,3,5},{2,4}}
{{1,4},{2,3,5}}
{{1,4,5},{2,3}}
{{1,5},{2,3,4}}
(End)
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
end:
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
MATHEMATICA
nn=20; p=Product[1+x^i/i!, {i, 1, nn}]; Drop[Range[0, nn]!CoefficientList[ Series[p, {x, 0, nn}], x], 1] (* Geoffrey Critzer, Sep 22 2012 *) a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)
PROG
(PARI) {my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017
CROSSREFS
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Number A(n,k) of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w; square array A(n,k), n>=0, k>=0, read by antidiagonals.
+10
27
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 4, 1, 0, 1, 1, 3, 10, 11, 1, 0, 1, 1, 3, 10, 23, 16, 1, 0, 1, 1, 3, 10, 47, 66, 42, 1, 0, 1, 1, 3, 10, 47, 126, 222, 64, 1, 0, 1, 1, 3, 10, 47, 246, 522, 561, 163, 1, 0, 1, 1, 3, 10, 47, 246, 882, 1821, 1647, 256, 1, 0
FORMULA
A(n,k) = Sum_{i=0..min(n,k)} A226874(n,i).
EXAMPLE
A(4,3) = 23: aaaa, aaab, aaba, aabb, aabc, aacb, abaa, abab, abac, abba, abca, acab, acba, baaa, baab, baac, baba, baca, bbaa, bcaa, caab, caba, cbaa.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 3, 3, 3, 3, 3, 3, ...
0, 1, 4, 10, 10, 10, 10, 10, ...
0, 1, 11, 23, 47, 47, 47, 47, ...
0, 1, 16, 66, 126, 246, 246, 246, ...
0, 1, 42, 222, 522, 882, 1602, 1602, ...
0, 1, 64, 561, 1821, 3921, 6441, 11481, ...
MAPLE
b:= proc(n, i, t) option remember;
`if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
end:
A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n-j, j, t-1]/j!, {j, i, n/t}]]; a[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)
CROSSREFS
Columns k=0-10 give: A000007, A000012, A027306, A092255, A092429, A226875, A226876, A226877, A226878, A226879, A226880.
Number T(n,k) of n-length words 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; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.
+10
19
1, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 10, 12, 24, 0, 1, 15, 50, 60, 120, 0, 1, 41, 180, 300, 360, 720, 0, 1, 63, 497, 1260, 2100, 2520, 5040, 0, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320, 0, 1, 255, 5154, 20916, 58464, 90720, 151200, 181440, 362880
COMMENTS
T(n,k) is the sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a multiset of size k.
EXAMPLE
T(4,2) = 10: aaab, aaba, aabb, abaa, abab, abba, baaa, baab, baba, bbaa.
T(4,3) = 12: aabc, aacb, abac, abca, acab, acba, baac, baca, bcaa, caab, caba, cbaa.
T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 3, 6;
0, 1, 10, 12, 24;
0, 1, 15, 50, 60, 120;
0, 1, 41, 180, 300, 360, 720;
0, 1, 63, 497, 1260, 2100, 2520, 5040;
0, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320;
...
MAPLE
b:= proc(n, i, t) option remember;
`if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
end:
T:= (n, k)-> `if`(n*k=0, `if`(n=k, 1, 0), n!*b(n, 1, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n, i) option remember; expand(
`if`(n=0, 1, `if`(i<1, 0, add(x^j*b(n-i*j, i-1)*
combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]]; t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from first Maple *)
PROG
(PARI)
T(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
{my(t=T(10)); for(n=1, #t, for(k=0, n-1, print1(polcoeff(t[n], k), ", ")); print)} \\ Andrew Howroyd, Dec 20 2017
CROSSREFS
Columns k=0-10 give: A000007, A057427, A226881, A226882, A226883, A226884, A226885, A226886, A226887, A226888, A226889.
Partition n labeled elements into sets of different sizes and order the sets.
+10
15
1, 1, 1, 7, 9, 31, 403, 757, 2873, 12607, 333051, 761377, 3699435, 16383121, 108710085, 4855474267, 13594184793, 76375572751, 388660153867, 2504206435681, 20148774553859, 1556349601444477, 5050276538344665, 33326552998257031, 186169293932977115, 1305062351972825281, 9600936552132048553, 106019265737746665727, 12708226588208611056333, 47376365554715905155127
COMMENTS
Also the number of matrices with n rows of nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to n and the column sums are distinct. Equivalently, the number of compositions of n into distinct parts where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once.
a(3) = 7:
[1] [1 0] [0 1] [1 0] [0 1] [0 1] [1 0]
[1] [1 0] [0 1] [0 1] [1 0] [1 0] [0 1]
[1] [0 1] [1 0] [1 0] [0 1] [1 0] [0 1].
3abc, 2ab1c, 1c2ab, 2ac1b, 1b2ac, 2bc1a, 1a2bc. (End)
FORMULA
"AGJ" (ordered, elements, labeled) transform of 1, 1, 1, 1, ...
MAPLE
b:= proc(n, i, p) option remember;
`if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p)+
`if`(i>n, 0, b(n-i, i-1, p+1)*binomial(n, i))))
end:
a:= n-> b(n$2, 0):
MATHEMATICA
f[list_]:=Apply[Multinomial, list]*Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 1, 30}]
b[n_, i_, p_] := b[n, i, p] = If[i*(i+1)/2<n, 0, If[n==0, p!, b[n, i-1, p] + If[i>n, 0, b[n-i, i-1, p+1]*Binomial[n, i]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)
PROG
(PARI) seq(n)=[subst(serlaplace(y^0*p), y, 1) | p <- Vec(serlaplace(prod(k=1, n, 1 + x^k*y/k! + O(x*x^n))))] \\ Andrew Howroyd, Sep 13 2018
Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size k; triangle T(n,k), n>=0, 0<=k<= A003056(n), read by rows.
+10
7
1, 0, 1, 0, 3, 0, 7, 3, 0, 31, 16, 0, 121, 125, 0, 831, 711, 60, 0, 5041, 5915, 525, 0, 42911, 46264, 6328, 0, 364561, 438681, 67788, 0, 3742453, 4371085, 753420, 12600, 0, 39916801, 49321745, 8924685, 166320, 0, 486891175, 588219523, 113501784, 2966040
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 3;
0, 7, 3;
0, 31, 16;
0, 121, 125;
0, 831, 711, 60;
0, 5041, 5915, 525;
0, 42911, 46264, 6328;
0, 364561, 438681, 67788;
0, 3742453, 4371085, 753420, 12600;
...
MAPLE
with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
select(x-> nops({x[]})=k, partition(n))):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*
combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);
MATHEMATICA
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[x^Sign[j]*b[n - i*j, i-1]*multinomial[n, Join[{n-i*j}, Table[i, {j}]]], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
Number of set partitions of [n] into two blocks with distinct sizes.
+10
3
3, 4, 15, 21, 63, 92, 255, 385, 1023, 1585, 4095, 6475, 16383, 26332, 65535, 106761, 262143, 431909, 1048575, 1744435, 4194303, 7036529, 16777215, 28354131, 67108863, 114159427, 268435455, 459312151, 1073741823, 1846943452, 4294967295, 7423131481, 17179869183
FORMULA
a(n) = n! * [x^n*y^2] Product_{n>=1} (1+y*x^n/n!).
MAPLE
b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n, i))))
end:
a:= n-> b(n$2, 2):
seq(a(n), n=3..40);
MATHEMATICA
Table[Sum[Binomial[n, i], {i, Floor[(n - 1)/2]}], {n, 3, 35}] (* Michael De Vlieger, Nov 15 2017 *)
PROG
(Magma) [(&+[Binomial(n, j): j in [1..Floor((n-1)/2)]]): n in [3..40]]; // G. C. Greubel, Jul 14 2024 (SageMath)
def A272514(n): return sum( binomial(n, j) for j in range(1, 1+((n-1)//2)))
Number of set partitions of [n] into three blocks with distinct sizes.
+10
2
60, 105, 448, 2016, 4980, 15675, 61644, 155155, 482573, 1733550, 4549808, 13890360, 48104628, 128949675, 392009140, 1322692581, 3607864403, 10929721440, 36245555284, 100109572875, 302709337515, 990788537700, 2763564406113, 8344789976616, 27039048750600
FORMULA
a(n) = n! * [x^n*y^3] Product_{n>=1} (1+y*x^n/n!).
MAPLE
b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n, i))))
end:
a:= n-> b(n$2, 3):
seq(a(n), n=6..40);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];
a[n_] := b[n, n, 3];
Number of set partitions of [n] into four blocks with distinct sizes.
+10
2
12600, 27720, 138600, 643500, 4408404, 12687675, 60780720, 238299880, 1295666424, 4208874756, 18840460800, 72351683460, 361100656224, 1228553894491, 5370616442928, 20605640103400, 97659853077800, 342942099783075, 1479570975628200, 5678915129142255
FORMULA
a(n) = n! * [x^n*y^4] Product_{n>=1} (1+y*x^n/n!).
MAPLE
b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n, i))))
end:
a:= n-> b(n$2, 4):
seq(a(n), n=10..40);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];
a[n_] := b[n, n, 4];
Number of set partitions of [n] into five blocks with distinct sizes.
+10
2
37837800, 100900800, 588107520, 2977294320, 20020160160, 164118754800, 635661248040, 3295178686800, 17741374681800, 95826446465904, 623399389674600, 2664090278249400, 13876038856379700, 71797074694745400, 375274098870636420, 2199911433079733100
FORMULA
a(n) = n! * [x^n*y^5] Product_{n>=1} (1+y*x^n/n!).
MAPLE
b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n, i))))
end:
a:= n-> b(n$2, 5):
seq(a(n), n=15..40);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];
a[n_] := b[n, n, 5];
Number of set partitions of [n] into six blocks with distinct sizes.
+10
2
2053230379200, 6453009763200, 43288940494800, 242418066770880, 1707999012720000, 12887361202716000, 144924867388501200, 620550897351184800, 4048435123506774000, 23424084614648718000, 161250104584826056800, 1013722794731975328000, 8616255173755280251200
FORMULA
a(n) = n! * [x^n*y^6] Product_{n>=1} (1+y*x^n/n!).
MAPLE
b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n, i))))
end:
a:= n-> b(n$2, 6):
seq(a(n), n=21..40);
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