A226874
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.
Original entry on oeis.org
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
Offset: 0
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;
...
Columns k=0-10 give:
A000007,
A057427,
A226881,
A226882,
A226883,
A226884,
A226885,
A226886,
A226887,
A226888,
A226889.
-
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$2)):
seq(T(n), n=0..12);
-
b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]]; t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from first Maple *)
-
T(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
{my(t=T(10)); for(n=1, #t, for(k=0, n-1, print1(polcoeff(t[n], k), ", ")); print)} \\ Andrew Howroyd, Dec 20 2017
A285849
Number T(n,k) of permutations of [n] with k ordered cycles such that equal-sized cycles are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 6, 1, 0, 6, 19, 18, 1, 0, 24, 100, 105, 40, 1, 0, 120, 508, 1005, 430, 75, 1, 0, 720, 3528, 6762, 6300, 1400, 126, 1, 0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1, 0, 40320, 219168, 558548, 706608, 431445, 105336, 9114, 288, 1
Offset: 0
T(3,1) = 2: (123), (132).
T(3,2) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 6, 1;
0, 6, 19, 18, 1;
0, 24, 100, 105, 40, 1;
0, 120, 508, 1005, 430, 75, 1;
0, 720, 3528, 6762, 6300, 1400, 126, 1;
0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1;
Columns k=0-10 give:
A000007,
A104150,
A285853,
A285854,
A285855,
A285856,
A285857,
A285858,
A285859,
A285860,
A285861.
-
b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
[multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)
A120774
Number of ordered set partitions of [n] where equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 1, 2, 8, 31, 147, 899, 5777, 41024, 322488, 2749325, 25118777, 245389896, 2554780438, 28009868787, 323746545433, 3933023224691, 49924332801387, 661988844566017, 9138403573970063, 131043199040556235, 1949750421507432009, 30031656711776544610
Offset: 0
A179233 begins 1; 1; 1 1; 6 1 1; 8 3 18 1 1 ... with row sums 1, 1 2 8 31 147 ...
a(3) = 8: 123, 1|23, 23|1, 2|13, 13|2, 3|12, 12|3, 1|2|3. - _Alois P. Heinz_, Apr 27 2017
-
b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
(p+n)!/n!, add(b(n-i*j, i-1, p+j)*combinat
[multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 27 2017
-
f[{x_,y_}]:= x!^y y!; Table[Total[Table[n!,{PartitionsP[n]}]/Apply[Times,Map[f,Map[Tally,Partitions[n]],{2}],2] * Apply[Multinomial,Map[Last,Map[Tally,Partitions[n]],{2}],2]],{n,0,20}] (* Geoffrey Critzer, Sep 29 2011 *)
Leading 1 inserted, definition simplified by
R. J. Mathar, Sep 28 2011
A285917
Number of ordered set partitions of [n] into two blocks such that equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 6, 11, 30, 52, 126, 219, 510, 896, 2046, 3632, 8190, 14666, 32766, 59099, 131070, 237832, 524286, 956196, 2097150, 3841586, 8388606, 15425136, 33554430, 61908562, 134217726, 248377154, 536870910, 996183062, 2147483646, 3994427099, 8589934590, 16013066072
Offset: 2
-
a:= n-> 2*add(binomial(n, k), k=1..n/2)-
`if`(n::even, 3/2*binomial(n, n/2), 0):
seq(a(n), n=2..40);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 11][n],
(9*(n-1)*(n-4)*a(n-1)+2*(3*n^2-16*n+6)*a(n-2)
-36*(n-2)*(n-4)*a(n-3)+8*(n-3)*(3*n-10)*a(n-4))
/((3*n-13)*n))
end:
seq(a(n), n=2..40);
-
a[n_] := 2*Sum[Binomial[n, k], {k, 1, n/2}] - If[EvenQ[n], 3/2*Binomial[n, n/2], 0];
Table[a[n], {n, 2, 40}] (* Jean-François Alcover, May 26 2018, from Maple *)
-
a(n) = 2*sum(k=1, n\2, binomial(n, k)) - if (!(n%2), 3*binomial(n, n/2)/2); \\ Michel Marcus, May 26 2018
A285918
Number of ordered set partitions of [n] into three blocks such that equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 18, 75, 420, 1218, 4242, 14563, 42930, 132528, 432960, 1250340, 3814629, 12073701, 35074482, 106044555, 331913202, 967193328, 2917846758, 9062084298, 26507831559, 79848170823, 246771097680, 723922691700, 2178960263415, 6709005218503, 19728686792637
Offset: 3
-
b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
[multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 4)
end:
a:= n-> coeff(b(n$2, 0), x, 3):
seq(a(n), n=3..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i-1, p+j]*multinomial[n, Join[{n - i*j}, Table[i, j] ] ]/j!^2, {j, 0, n/i}]], {x, 0, 4}];
a[n_] := Coefficient[b[n, n, 0], x, 3];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)
A285919
Number of ordered set partitions of [n] into four blocks such that equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 40, 350, 3080, 17129, 82488, 464650, 1901680, 8357426, 35701952, 159721016, 627687060, 2642405289, 10712590392, 45568675202, 178738923440, 736145997686, 2946913512648, 12311241803256, 48275516469180, 197284995875314, 786939537437440, 3254422571085400
Offset: 4
-
b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
[multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 5)
end:
a:= n-> coeff(b(n$2, 0), x, 4):
seq(a(n), n=4..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i - 1, p + j]*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2, {j, 0, n/i}]], {x, 0, 5}];
a[n_] := Coefficient[b[n, n, 0], x, 4];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)
A285920
Number of ordered set partitions of [n] into five blocks such that equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 75, 1225, 15750, 152355, 1049895, 8130925, 51541050, 305751160, 1721589870, 10370592050, 54481859250, 292852136335, 1539187989915, 8149972381105, 43456591157700, 220640087499230, 1133640238666320, 5822084961637780, 29811110400741780, 154396823960132126
Offset: 5
-
b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
[multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 6)
end:
a:= n-> coeff(b(n$2, 0), x, 5):
seq(a(n), n=5..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i - 1, p + j]*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2, {j, 0, n/i}]], {x, 0, 6}];
a[n_] := Coefficient[b[n, n, 0], x, 5];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)
A285921
Number of ordered set partitions of [n] into six blocks such that equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 126, 3486, 63756, 954387, 9628542, 97141022, 886634892, 7048863822, 53483658228, 397751490318, 2858731936788, 19510233553063, 130084038669798, 844004265958794, 5657554841332464, 35647504639822614, 227439073802247384, 1425548351910315534, 8934412155886521480
Offset: 6
-
b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
[multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 7)
end:
a:= n-> coeff(b(n$2, 0), x, 6):
seq(a(n), n=6..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i - 1, p + j]*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2, {j, 0, n/i}]], {x, 0, 7}];
a[n_] := Coefficient[b[n, n, 0], x, 6];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)
A285922
Number of ordered set partitions of [n] into seven blocks such that equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 196, 8526, 217560, 4635939, 67454772, 877414538, 10742461730, 113528563148, 1132899916148, 10494458555126, 96114856972680, 831333224017303, 7005224782844764, 56197005110455286, 453234116137501160, 3555422918860518398, 27541742188014185824
Offset: 7
-
b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
[multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 8)
end:
a:= n-> coeff(b(n$2, 0), x, 7):
seq(a(n), n=7..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i - 1, p + j]*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2, {j, 0, n/i}]], {x, 0, 8}] ;
a[n_] := Coefficient[b[n, n, 0], x, 7];
Table[a[n], {n, 7, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)
A285923
Number of ordered set partitions of [n] into eight blocks such that equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 288, 18600, 649440, 18650346, 378728064, 6346968056, 99768480240, 1370094506209, 17452476893280, 204026690329800, 2291047776886752, 24663963563727574, 256637317406331648, 2540192740448641960, 24558666993552144288, 233835181800425532162
Offset: 8
-
b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
[multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 9)
end:
a:= n-> coeff(b(n$2, 0), x, 8):
seq(a(n), n=8..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i - 1, p + j]*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2, {j, 0, n/i}]], {x, 0, 9}];
a[n_] := Coefficient[b[n, n, 0], x, 8];
Table[a[n], {n, 8, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)
Showing 1-10 of 13 results.
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