The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A092582 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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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.
(history; published version)

#72 by Alois P. Heinz at Fri Apr 28 20:08:08 EDT 2023

STATUS

editing

approved

#71 by Alois P. Heinz at Fri Apr 28 20:06:59 EDT 2023

FORMULA

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

CROSSREFS

Cf. A000522.

STATUS

approved

editing

#70 by Alois P. Heinz at Fri Apr 28 19:55:37 EDT 2023

STATUS

editing

approved

#69 by Alois P. Heinz at Fri Apr 28 19:55:34 EDT 2023

CROSSREFS

Cf. A000166, A002627, A055596, A092580, A092956, A130477.

STATUS

approved

editing

#68 by Peter Luschny at Wed Sep 07 04:08:02 EDT 2022

STATUS

reviewed

approved

#67 by Joerg Arndt at Wed Sep 07 03:16:05 EDT 2022

STATUS

proposed

reviewed

#66 by Michel Marcus at Tue Sep 06 02:45:12 EDT 2022

STATUS

editing

proposed

#65 by Michel Marcus at Tue Sep 06 02:45:06 EDT 2022

LINKS

Olivier Bodini, Antoine Genitrini, and Mehdi Naima, <a href="https://arxiv.org/abs/1808.08376">Ranked Schröder Trees</a>, arXiv:1808.08376 [cs.DS], 2018.

Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, <a href="https://hal.archives-ouvertes.fr/hal-02865198">Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study</a>, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.

Colin Defant, and James Propp, <a href="https://arxiv.org/abs/2002.07144">Quantifying Noninvertibility in Discrete Dynamical Systems</a>, arXiv:2002.07144 [math.CO], 2020.

E. Emeric Deutsch and W. P. Johnson, <a href="http://www.jstor.org/stable/3219101">Create your own permutation statistics</a>, Math. Mag., 77, 130-134, 2004.

STATUS

proposed

editing

#64 by G. C. Greubel at Tue Sep 06 01:29:03 EDT 2022

STATUS

editing

proposed

#63 by G. C. Greubel at Tue Sep 06 01:28:48 EDT 2022

FORMULA

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)

PROG

(Magma)

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

(SageMath)

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

CROSSREFS

Cf. A000166.

Cf. A000166, A002627, A055596, A092956, A130477.

Cf. A000027, A001710, A001715, A001720, A001725, A005563.

STATUS

approved

editing