A089759 -id:A089759 - cp's OEIS frontend

A372307 Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 9, 10, 1, 0, 1, 1, 44, 297, 56, 1, 0, 1, 1, 265, 13756, 13833, 346, 1, 0, 1, 1, 1854, 925705, 6699824, 748521, 2252, 1, 0, 1, 1, 14833, 85394646, 5691917785, 3993445276, 44127009, 15184, 1, 0, 1

Offset: 0

Jeremy Tan, Apr 26 2024

A deck has k suits of n cards each. The deck is shuffled and dealt into k hands of n cards each. A match occurs for every card in the i-th hand of suit i. T(n,k) is the number of ways of achieving no matches. The probability of no matches is T(n,k)/((n*k)!/n!^k).

T(n,k) is the maximal number of totally mixed Nash equilibria in games of k players, each with n+1 pure options.

			Square array T(n,k) begins:
  1, 1, 1,      1,            1,                   1, ...
  1, 0, 1,      2,            9,                  44, ...
  1, 0, 1,     10,          297,               13756, ...
  1, 0, 1,     56,        13833,             6699824, ...
  1, 0, 1,    346,       748521,          3993445276, ...
  1, 0, 1,   2252,     44127009,       2671644472544, ...
  1, 0, 1,  15184,   2750141241,    1926172117389136, ...
  1, 0, 1, 104960, 178218782793, 1463447061709156064, ...

Jeremy Tan, Antidiagonals n = 0..32, flattened
Shalosh B. Ekhad, Christoph Koutschan and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
S. Even and J. Gillis, Derangements and Laguerre polynomials, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 79, Issue 1, January 1976, pp. 135-143.
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), pp. 107-118.
Jeremy Tan, Python program
Raimundas Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv:1401.5400 [math.CO], 2014.
Raimundas Vidunas, Counting derangements and Nash equilibria, Ann. Comb. 21, No. 1, 131-152 (2017).
Index entries for sequences related to card matching

Rows 0-5 give A000012, A000166, A000459, A059073, A059074, A123297.
Columns 0-4 give A000012, A000007, A000012, A000172, A371252.
Main diagonal gives A375778.

A:= (n, k)-> (-1)^(n*k)*int(exp(-x)*orthopoly[L](n, x)^k, x=0..infinity):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 27 2024

Table[Abs[Integrate[Exp[-x] LaguerreL[n, x]^(s-n), {x, 0, Infinity}]], {s, 0, 9}, {n, 0, s}] // Flatten

Python
```
# See link.
```

T(n,k) = (-1)^(n*k) * Integral_{x=0..oo} exp(-x)*L_n(x)^k dx, where L_n(x) is the Laguerre polynomial of degree n (Even and Gillis).

T(n,k) ~ A089759(n,k)/exp(n).

A120666 Triangle read by rows: T(n, k) = (n*k)!/(n!)^k.

Original entry on oeis.org

1, 1, 6, 1, 20, 1680, 1, 70, 34650, 63063000, 1, 252, 756756, 11732745024, 623360743125120, 1, 924, 17153136, 2308743493056, 1370874167589326400, 2670177736637149247308800, 1, 3432, 399072960, 472518347558400, 3177459078523411968000, 85722533226982363751829504000, 7363615666157189603982585462030336000

Offset: 1

Roger L. Bagula, Aug 11 2006

T(m,n) is the number of ways to distribute n*m different toys among m different kids so that each kid gets exactly n toys. For example, with n=3 and m=2, the 6 different toys, t1, t2, t3, t4, t5 and t6, can be distributed in exactly 20 ways among the 2 kids, k1 and k2, since there are C(6,3)=20 ways to choose the three toys for k1, with the other three toys going to k2. The proof for the general case is based on the identity C(n*m,n)*C(n*m-n,n)*...*C(n*m-n*(m-1),n) = (n*m)!/(n!)^m. - Dennis P. Walsh, Apr 12 2018

			Triangle begins:
  1;
  1,   6;
  1,  20,   1680;
  1,  70,  34650,    63063000;
  1, 252, 756756, 11732745024, 623360743125120;

Seiichi Manyama, Rows n = 1..26, flattened
Eric Weisstein's World of Mathematics, Macdonald's Constant-Term Conjecture

Cf. A000984, A006480, A034841, A089759, A187783.

[Factorial(n*k)/(Factorial(n))^k: k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 26 2022

T:= (m, n)-> (n*m)!/(m!)^n:
seq(seq(T(m, n), n=1..m), m=1..7);  # Alois P. Heinz, Apr 12 2018

Table[(n*k)!/(n!)^k, {n,10}, {k,n}]//Flatten

def A120666(n,k): return gamma(n*k+1)/(factorial(n))^k
flatten([[A120666(n,k) for k in range(1,n+1)] for n in range(1,11)]) # G. C. Greubel, Dec 26 2022

T(n, k) = (k*n)!/(n!)^k.

Edited by N. J. A. Sloane, Jun 17 2007
Offset corrected by Alois P. Heinz, Apr 12 2018
New name using formula by Joerg Arndt, Apr 15 2018

A375694 Number A(n,k) of multiset permutations of {{1}^k, {2}^k, ..., {n}^k} with no fixed k-tuple {j}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 5, 2, 0, 1, 0, 19, 74, 9, 0, 1, 0, 69, 1622, 2193, 44, 0, 1, 0, 251, 34442, 362997, 101644, 265, 0, 1, 0, 923, 756002, 62924817, 166336604, 6840085, 1854, 0, 1, 0, 3431, 17150366, 11729719509, 305225265804, 136221590695, 630985830, 14833, 0

Offset: 0

Alois P. Heinz, Aug 24 2024

			A(2,2) = 5: 1212, 1221, 2112, 2121, 2211.
A(2,3) = 19: 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.
A(3,2) = 74: 121323, 121332, 122313, 122331, 123123, 123132, 123213, 123231, 123312, 123321, 131223, 131232, 131322, 132123, 132132, 132312, 132321, 133122, 133212, 133221, 211323, 211332, 212313, 212331, 213123, 213132, 213213, 213231, 213312, 213321, 221313, 221331, 223113, 223131, 223311, 231123, 231132, 231213, 231231, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311223, 311232, 311322, 312123, 312132, 312312, 312321, 313122, 313212, 313221, 321123, 321132, 321213, 321231, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121.
A(4,1) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Square array A(n,k) begins:
  1,  1,      1,         1,            1,               1, ...
  0,  0,      0,         0,            0,               0, ...
  0,  1,      5,        19,           69,             251, ...
  0,  2,     74,      1622,        34442,          756002, ...
  0,  9,   2193,    362997,     62924817,     11729719509, ...
  0, 44, 101644, 166336604, 305225265804, 623302086965044, ...

Alois P. Heinz, Antidiagonals n = 0..53, flattened

Columns k=0-2 give: A000007, A000166, A374980.
Rows n=0-2 give: A000012, A000004, A030662.
Main diagonal gives A375693.
Cf. A060538, A089759, A187783, A372307.

A:= (n, k)-> add((-1)^(n-j)*binomial(n, j)*(k*j)!/k!^j, j=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(k*j)!/k!^j.

A198801 Number of closed paths of length 19n whose steps are 19th roots of unity.

Original entry on oeis.org

1, 121645100408832000, 997586474354936812896742294502400000000, 66507379539349211518364492838558005847493108039680000000000000, 11256716378122801825351385824819232115042452248916289339300576523719750000000000000000

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

Equivalently, the number of paths of length 19n in Z^19 from {0}^19 to {n}^19. - Andrew Howroyd, Nov 01 2018

Andrew Howroyd, Table of n, a(n) for n = 0..40
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Row n=19 of A187783, column k=19 of A089759.

a(n) = (19*n)!/(n!)^19; \\ Andrew Howroyd, Nov 01 2018

a(n) = (19*n)!/(n!)^19. - Andrew Howroyd, Nov 01 2018

Sequence redefined and a(2)-a(4) from Andrew Howroyd, Nov 01 2018

A198803 Number of closed paths of length 17n whose steps are 17th roots of unity.

Original entry on oeis.org

1, 355687428096000, 2252447502438386084347676160000000, 91637618063484032681381970173925718228992000000000000, 8528384964488882787308232082310780143738202829970606470279000000000000000

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

Equivalently, the number of paths of length 17n in Z^17 from {0}^17 to {n}^17. - Andrew Howroyd, Nov 01 2018

Andrew Howroyd, Table of n, a(n) for n = 0..40
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Row n=17 of A187783, column k=17 of A089759.

a(n) = (17*n)!/(n!)^17; \\ Andrew Howroyd, Nov 01 2018

a(n) = (17*n)!/(n!)^17. - Andrew Howroyd, Nov 01 2018

Sequence redefined and a(2)-a(4) from Andrew Howroyd, Nov 01 2018

A198809 Number of closed paths of length 11n whose steps are 11th roots of unity.

Original entry on oeis.org

1, 39916800, 548828480360160000, 23934366266775567482880000000, 1746930746117010628955362040959500000000, 170878335353097656943918169452451079403744627916800, 20193738534370392855946567010492898163504440783192016158720000

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

Equivalently, the number of paths of length 11n in Z^11 from {0}^11 to {n}^11. - Andrew Howroyd, Nov 01 2018

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Row n=11 of A187783, column k=11 of A089759.

a(n) = (11*n)!/(n!)^11; \\ Andrew Howroyd, Nov 01 2018

a(n) = (11*n)!/(n!)^11. - Andrew Howroyd, Nov 01 2018

Sequence redefined and a(2)-a(6) from Andrew Howroyd, Nov 01 2018

A198807 Number of closed paths of length 13n whose steps are 13th roots of unity.

Original entry on oeis.org

1, 6227020800, 49229914688306352000000, 1561776277448122046153927884800000000, 92024242230271040357108320801872044844750000000000, 7708574168669332219803079339976372645861971547841327593737420800

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

Equivalently, the number of paths of length 13n in Z^13 from {0}^13 to {n}^13. - Andrew Howroyd, Nov 01 2018

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Row n=13 of A187783, column k=13 of A089759.

a(n) = (13*n)!/(n!)^13; \\ Andrew Howroyd, Nov 01 2018

a(n) = (13*n)!/(n!)^13. - Andrew Howroyd, Nov 01 2018

Sequence redefined and a(2)-a(5) from Andrew Howroyd, Nov 01 2018

cp's OEIS Frontend

A372307 Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.

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A120666 Triangle read by rows: T(n, k) = (n*k)!/(n!)^k.

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A375694 Number A(n,k) of multiset permutations of {{1}^k, {2}^k, ..., {n}^k} with no fixed k-tuple {j}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

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A198801 Number of closed paths of length 19n whose steps are 19th roots of unity.

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PARI

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A198803 Number of closed paths of length 17n whose steps are 17th roots of unity.

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Crossrefs

Programs

PARI

Formula

Extensions

A198809 Number of closed paths of length 11n whose steps are 11th roots of unity.

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Keywords

Comments

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Crossrefs

Programs

PARI

Formula

Extensions

A198807 Number of closed paths of length 13n whose steps are 13th roots of unity.

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Author

Keywords

Comments

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Crossrefs

Programs