A173333 -id:A173333 - cp's OEIS frontend

A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

1, 1, 1, 3, 2, 1, 12, 8, 3, 1, 60, 40, 15, 4, 1, 360, 240, 90, 24, 5, 1, 2520, 1680, 630, 168, 35, 6, 1, 20160, 13440, 5040, 1344, 280, 48, 7, 1, 181440, 120960, 45360, 12096, 2520, 432, 63, 8, 1, 1814400, 1209600, 453600, 120960, 25200, 4320, 630, 80, 9, 1

Offset: 1

Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004

Row sums are the factorial numbers (A000142). First column is A001710.
T(n,k) = number of permutations of [n] in which 1,2,...,k is a subsequence but 1,2,...,k,k+1 is not. Example: T(4,2)=8 because 1324, 1342, 1432, 4132, 3124, 3142, 3412 and 4312, are the only permutations of [4] in which 12 is a subsequence but 123 is not. - Emeric Deutsch, Nov 12 2004
T(n,k) is the number of deco polyominoes of height n with k cells in the last column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column). - Emeric Deutsch, Jan 06 2005
T(n,k) is the number of permutations p of [n] for which the smallest i such that p(i)Emeric Deutsch, Feb 23 2008
Adding columns 2,4,6,... one obtains the derangement numbers 0,1,2,9,44,... (A000166). See the Bona reference (p. 118, Exercises 41,42). - Emeric Deutsch, Feb 23 2008
Matrix inverse of A128227*A154990. - Mats Granvik, Feb 08 2009
Differences in the columns of A173333 which counts the n-permutations with an initial ascending run of length at least k. - Geoffrey Critzer, Jun 18 2017
The triangle with each row reversed is A130477. - Michael Somos, Jun 25 2017

			T(4,3) = 3 because 1243, 1342 and 2341 are the only permutations of [4] having length of first run equal to 3.
     1;
     1,    1;
     3,    2,   1;
    12,    8,   3,   1;
    60,   40,  15,   4,  1;
   360,  240,  90,  24,  5,  1;
  2520, 1680, 630, 168, 35,  6,  1;
  ...

M. Bona, Combinatorics of Permutations, Chapman&Hall/CRC, Boca Raton, Florida, 2004.

Alois P. Heinz, Rows n = 1..141, flattened
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Colin Defant and James Propp, Quantifying Noninvertibility in Discrete Dynamical Systems, arXiv:2002.07144 [math.CO], 2020.
Emeric Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.

Cf. A002260. - Gary W. Adamson, Feb 22 2012
Cf. A000166, A002627, A055596, A092580, A092956, A130477.
Cf. A000027, A001710, A001715, A001720, A001725, A005563.
Cf. A000522.

Flat(List([1..11],n->Concatenation([1],List([1..n-1],k->Factorial(n)*k/Factorial(k+1))))); # Muniru A Asiru, Jun 10 2018

A092582:= func< n,k | k eq n select 1 else k*Factorial(n)/Factorial(k+1) >;
[A092582(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 06 2022

Drop[Drop[Abs[Map[Select[#, # < 0 &] &, Map[Differences, nn = 10; Range[0, nn]! CoefficientList[Series[(Exp[y x] - 1)/(1 - x), {x, 0, nn}], {x, y}]]]], 1], -1] // Grid (* Geoffrey Critzer, Jun 18 2017 *)

{T(n, k) = if( n<1 || k>n, 0, k==n, 1, n! * k /(k+1)!)}; /* Michael Somos, Jun 25 2017 */

def A092582(n,k): return 1 if (k==n) else k*factorial(n)/factorial(k+1)
flatten([[A092582(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Sep 06 2022

T(n, k) = n!*k/(k+1)! for k

Inverse of:

-1, 1;

-1, -2, 1;

-1, -2, -3, 1;

-1, -2, -3, -4, 1;

... where A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Feb 22 2012

T(2n,n) = A092956(n-1) for n>0. - Alois P. Heinz, Jun 19 2017

From Alois P. Heinz, Dec 17 2021: (Start)

Sum_{k=1..n} k * T(n,k) = A002627(n).

|Sum_{k=1..n} (-1)^k * T(n,k)| = A055596(n) for n>=1. (End)

From G. C. Greubel, Sep 06 2022: (Start)

T(n, 1) = A001710(n).

T(n, 2) = 2*A001715(n) + [n=2]/3, n >= 2.

T(n, 3) = 3*A001720(n) + [n=3]/4, n >= 3.

T(n, 4) = 4*A001725(n) + [n=4]/5, n >= 4.

T(n, n-1) = A000027(n-1).

T(n, n-2) = A005563(n-1), n >= 3. (End)

Sum_{k=0..n} (k+1) * T(n,k) = A000522(n). - Alois P. Heinz, Apr 28 2023

A119741 A008279, with the first and last of each row removed.

Original entry on oeis.org

2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 6, 30, 120, 360, 720, 7, 42, 210, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800

Offset: 2

Lekraj Beedassy, Jul 29 2006

Triangle read by rows: T(n,k) (n>=2, k=1..n-1) is the number of topologies t on n points having exactly k+2 open sets such that t contains exactly one open set of size m for each m in {0,1,2,...,s,n} where s is the size of the largest proper open set in t. - N. J. A. Sloane, Jan 29 2016 [clarified by Geoffrey Critzer, Feb 19 2017]

			Triangle begins:
   2;
   3,  6;
   4, 12,  24;
   5, 20,  60,  120;
   6, 30, 120,  360,   720;
   7, 42, 210,  840,  2520,   5040;
   8, 56, 336, 1680,  6720,  20160,  40320;
   9, 72, 504, 3024, 15120,  60480, 181440,  362880;
  10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Row sums give A038156.
Triangles in this series: A268216, A268217, A268221, A268222, A268223.
Cf. A008279, A014631, A282507.

T:= (n, k)-> n!/(n-k)!:
seq(seq(T(n,k), k=1..n-1), n=2..11);  # Alois P. Heinz, Aug 22 2025

Table[FactorialPower[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 21 2020 *)

a(n) = (A003057(n))!/(A004736(n))! = (A002260(n))!*(A014410(n)).

T(n,k) = A173333(n+1,n-k+1), 1<=k<=n. - Reinhard Zumkeller, Feb 19 2010

Edited by Don Reble, Aug 01 2006

A159083 Products of 7 consecutive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 17297280, 32432400, 57657600, 98017920, 160392960, 253955520, 390700800, 586051200, 859541760, 1235591280, 1744364160, 2422728000, 3315312000, 4475671200, 5967561600, 7866331200

Offset: 0

Zerinvary Lajos, Apr 05 2009

Michael De Vlieger and Harvey P. Dale, Table of n, a(n) for n = 0..10000 (first 1000 terms by Harvey P. Dale.)
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A052762, A052787, A053625, A173333.

Equals A008279(n,7) (for n>=7).

I:=[0,0,0,0,0,0,0,5040]; [n le 8 select I[n] else 8*Self(n-1) - 28*Self(n-2) +56*Self(n-3) -70*Self(n-4) +56*Self(n-5) -28*Self(n-6) +8*Self(n-7) -Self(n-8): n in [1..30]]; // G. C. Greubel, Jun 28 2018

G(x):=x^7*exp(x): f[0]:=G(x): for n from 1 to 36 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..33);

Table[Times@@(n+Range[0,6]),{n,-6,25}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,0,0,0,0,0,0,5040},30] (* Harvey P. Dale, Apr 07 2018 *)

my(x='x+O('x^30)); concat([0,0,0,0,0,0,0], Vec(5040*x^7/(1-x)^8)) \\ G. C. Greubel, Jun 28 2018

E.g.f.: x^7*exp(x).
For n>=8: a(n) = A173333(n,n-7). - Reinhard Zumkeller, Feb 19 2010
G.f.: 5040*x^7/(1-x)^8. - Colin Barker, Mar 27 2012
From Amiram Eldar, Mar 08 2022: (Start)
a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6) = n!/(n-7)!.
Sum_{n>=7} 1/a(n) = 1/4320.
Sum_{n>=7} (-1)^(n+1)/a(n) = 4*log(2)/45 - 1327/21600. (End)

A162995 A scaled version of triangle A162990.

Original entry on oeis.org

1, 3, 1, 12, 4, 1, 60, 20, 5, 1, 360, 120, 30, 6, 1, 2520, 840, 210, 42, 7, 1, 20160, 6720, 1680, 336, 56, 8, 1, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Johannes W. Meijer, Jul 27 2009

We get this scaled version of triangle A162990 by dividing the coefficients in the left hand columns by their 'top-values' and then taking the square root.
T(n,k) = A173333(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Feb 19 2010
T(n,k) = A094587(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

			The first few rows of the triangle are:
[1]
[3, 1]
[12, 4, 1]
[60, 20, 5, 1]

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A094587.
A056542(n) equals the row sums for n>=1.
A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431 are related to the left hand columns.
A000012, A009056, A002378, A007531, A052762, A052787, A053625 and A159083 are related to the right hand columns.

a162995 n k = a162995_tabl !! (n-1) !! (k-1)
a162995_row n = a162995_tabl !! (n-1)
a162995_tabl = map fst $ iterate f ([1], 3)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

a := proc(n, m): (n+1)!/(m+1)! end: seq(seq(a(n, m), m=1..n), n=1..9); # Johannes W. Meijer, revised Nov 23 2012

Table[(n+1)!/(m+1)!, {n, 10}, {m, n}] (* Paolo Xausa, Mar 31 2024 *)

a(n,m) = (n+1)!/(m+1)! for n = 1, 2, 3, ..., and m = 1, 2, ..., n.

A213936 Number triangle with entry a(n,k), n>=1, m=1, 2, ..., n, giving the number of representative necklaces with n beads (C_n symmetry) corresponding to the color multinomial c[1]^kc[2]...*c[n+1-k].

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 24, 12, 4, 1, 1, 120, 60, 20, 5, 1, 1, 720, 360, 120, 30, 6, 1, 1, 5040, 2520, 840, 210, 42, 7, 1, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 1

Offset: 1

Wolfdieter Lang, Jul 10 2012

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.

			n\k      1       2      3      4     5    6   7  8  9 10 ...
1        1
2        1       1
3        2       1      1
4        6       3      1      1
5       24      12      4      1     1
6      120      60     20      5     1    1
7      720     360    120     30     6    1   1
8     5040    2520    840    210    42    7   1  1
9    40320   20160   6720   1680   336   56   8  1  1
10  362880  181440  60480  15120  3024  504  72  9  1  1 ...
a(4,3) = 1 because  the partition is [3,1], the color signature (exponentiation) c[.]^3 c[.]^1, and the one representative necklace (we use j for color c[j] here) is: cyclic(1112).
a(4,2) = 3  because  the partition is [2,1^2], the color signature c[.]^2 c[.] c[.], and the three representative necklaces are: cyclic(1123), cyclic(1132) and cyclic(1213).
a(5,3) = 4  because the color signature is  c[.]^3 c[.] c[.]  (from the partition [3,1^2]). and the four representative necklaces are 11123, 11132, 11213 and 11312, all taken cyclically.

Cf. A212359, A213937 (row sums). For columns and diagonals see the links under A173333 (after an additional 1 has been supplied for each columns).

a(n,n)=1, a(n,k) = (n-1)!/k! if 1 <= k < n, else 0.
See also A212359 with a link for the formula for general partitions.
a(n,k) = A173333(n-1,k), 1 <= k < n.

A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 6, 120, 60, 20, 5, 1, 30, 10, 720, 360, 120, 30, 6, 1, 180, 60, 15, 20, 90

Offset: 1

Alford Arnold, Mar 27 2008

Multinomials count permutations of multisets and also paths in lattices; for example, there are six paths (from null to full) through the lattice of divisors for signature 36: 2233 2323 2332 3223 3232 and 3322.

			a(11) is six because the eleventh least prime signature in source format is 36 the signature for partition 2+2 the ninth partition and A036038(9) = 6.
The tables begin:
1.......2.......6.......24......120.....720....5040.....40320......362880
........1.......3.......12.......60.....360....2520.....20160......181440
................1.......4........20.....120.....840......6720.......60480
........................1........5.......30.....210......1680.......15120
.. ..............................1........6......42......336........3024
..........................................1.......7.......56.........504
..................................................1........8..........72
...........................................................1...........9
.......................................................................1
........................6........30.....180....1260....10080........90720
.................................10......60.....420.....3360........30240
...

Cf. A000142 A001710 A005651 (column sums) A025487 A053445.

Cf. A173333. [From Reinhard Zumkeller, Feb 19 2010]

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
  n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
m
1         1
2         2      1
3         6      3     1
4        24     12     4     1
5       120     60    20     5    1
6       720    360   120    30    6   1
7      5040   2520   840   210   42   7   1
8     40320  20160  6720  1680  336  56   8   1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Tilman Piesk, Triangle rows m=0..25, flattened
Tilman Piesk, Illustration of the multisets for m=0..6

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A376582 Triangle of generalized Stirling numbers.

Original entry on oeis.org

1, 5, 1, 26, 7, 1, 154, 47, 9, 1, 1044, 342, 74, 11, 1, 8028, 2754, 638, 107, 13, 1, 69264, 24552, 5944, 1066, 146, 15, 1, 663696, 241128, 60216, 11274, 1650, 191, 17, 1, 6999840, 2592720, 662640, 127860, 19524, 2414, 242, 19, 1, 80627040, 30334320, 7893840, 1557660, 245004, 31594, 3382, 299, 21, 1

Offset: 0

Igor Victorovich Statsenko, Sep 29 2024

			Triangle starts:
[0]       1;
[1]       5,       1;
[2]      26,       7,       1;
[3]     154,      47,       9,        1;
[4]    1044,     342,      74,       11,       1;
[5]    8028,    2754,     638,      107,      13,     1;
[6]   69264,   24552,    5944,     1066,     146,    15,    1;
[7]  663696,  241128,   60216,    11274,    1650,   191,   17,    1;

Column k: A001705 (k=0), A001711 (k=1), A001716 (k=2), A001721 (k=3), A051524 (k=4), A051545 (k=5), A051560 (k=6).

Cf. A094587 and A173333 for m=0.

T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k): m:=1: seq(seq(T(m,n,k), k=0..n), n=0..10);

T(m,n,k) = Sum_{i=0..n-k} Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!, for m=1.

A360587 a(n) is the least positive integer k such that k(k+1)...*(k+n-1) does not contain the digit 2, or -1 if there is no such k.

Original entry on oeis.org

1, 2, 1, 3, 7, 2, 1, 3, 3, 2, 1, 1, 3, 2, 1, 5, 10, 10, 10, 17, 4, 8, 38, 38, 19, 17, 2, 1, 1, 3

Offset: 1

Robert Israel, Feb 12 2023

a(n) is the least positive integer k such that (k+n-1)!/(k-1)! does not contain the digit 2, or -1 if there is no such k.
a(32) = 13.
Conjecture: a(n) = -1 for n = 31 and all n >= 33.

			a(4) = 3 because 3*4*5*6 = 360 does not contain the digit 2, while 1*2*3*4 = 24 and 2*3*4*5 = 120 do.

Cf. A173333.

f:= proc(n) local k,t;
t:= n!;
for k from 1 to 100000 do
   if not member(2,convert(t,base,10)) then return k fi;
   t:= t*(n+k)/k;
od:
-1
end proc:
map(f, [$1..32]);

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A053625 Product of 6 consecutive integers.

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A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

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A119741 A008279, with the first and last of each row removed.

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A159083 Products of 7 consecutive integers.

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A162995 A scaled version of triangle A162990.

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A213936 Number triangle with entry a(n,k), n>=1, m=1, 2, ..., n, giving the number of representative necklaces with n beads (C_n symmetry) corresponding to the color multinomial c[1]^k*c[2]*...*c[n+1-k].

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A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

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A213936 Number triangle with entry a(n,k), n>=1, m=1, 2, ..., n, giving the number of representative necklaces with n beads (C_n symmetry) corresponding to the color multinomial c[1]^kc[2]...*c[n+1-k].

A360587 a(n) is the least positive integer k such that k(k+1)...*(k+n-1) does not contain the digit 2, or -1 if there is no such k.